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ISSN 2074-1871 Уфимский математический журнал. Том 15. № 1 (2023). С. 56-121.
INEXISTENCE OF NON-PRODUCT HESSIAN RANK 1 AFFINELY HOMOGENEOUS HYPERSURFACES Hn c Rn+1
IN DIMENSION n ^ 5
J. MERKER
Abstract. Equivalences under the affine group Aff(R3) of constant Hessian rank 1 surfaces c R3, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. In particular, homogeneous models and algebras of differential invariants in various branches were fully understood.
Then what is about higher dimensions? We consider hvpersurfaces Hn C Rra+1 graphed as {u = F(x]_,..., xn)} whose Hessian matrix (FXiXj), a relative affine invariant, is similarly 1
Complete explorations were done by the author on a computer in dimensions n = 2, 3, 4, 5, 6, 7. The first, expected outcome, was a complete classification of homogeneous models in dimensions n = 2, 3, 4 (forthcoming article, case n = 2 already known). The second, unexpected outcome, was that in dimensions n = 5, 6, 7, there are no aflinely homogenous models except those that are aflinely equivalent to a product of Rm with a
2, 3, 4
The present article establishes such a non-existence result in every dimension n ^ 5, based on the production of a normal form for {u = F(x-\_,... ,xn)}, under Aff(Rra+1) up to order ^ n + 5, valid in any dimension n ^ 2.
Keywords: Affine homogeneity, Normal forms, tangential vector fields.
Mathematics Subject Classification: 53A55, 53B25, 53A15, 53A04, 53A05, 58K50, 16W22, 14R20, 22E05, 35B06.
1. Introduction
Let n ^ 1 x = (x1,..., xn) e Rn, u e R and y = (y1,... ,yn) e R^ v e R. We consider equivalences of local analytic hvpersurfaces Hn C Kn C R™+j;1 graphed as:
u = F(x1,...,xn) and v = G(y1,... ,yn),
under affine transformations Rra+1 —> Rra+1:
y1 = 0,1,1 + ■ ■ ■ + ai,n xn + u + T1,
yn &n,1 + • • • + 0>n,n Xn + bn ^ + T^ni
V = Ci Xi +----+ cn xn + du + To,
J. Merker, Inexistence of non-product Hessian rank 1 affinely homogeneous hypersurfaces Hn c rn+1 in dimension n > 5. © Merker J. 2023.
The reported study by J. Merker was funded in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583, and by the Norwegian Financial Mechanism 2014-2021 via the project registration number 2019/34/H/ST1/0Û636. Submitted February 1, 2022.
where the (n + 1) x (n + 1) linear-part (" ^) matrix is invertible, i.e., belongs to GL(n + 1, R). The collection of all these transformations is the Lie transformation group Aff(Rra+1), The Lie algebra aff(Rra+1) of Aff(Rra+1) consists of the vector fields:
d d d L = Ti — + ••• + Tn — + To — OX1 oxn ou
+ (^1,1 +-----+ A1,n xn + B1 uj
d
+
dx1
+ (An,1 X1 +-----+ An,n xn + Bn u) +
V / oxn
+ X1 +-----+ Cn xn + Du^
A fixed hypersurface Hn C Rra+1 possesses an affine symmetry group, which is a local Lie group,for background see [1, Chap, 3]:
Sym(H) := e Aff(Rra+1): V(H) C H},
where " C " is understood up to shrinking H, and where the transfermations ^ are close to the identity. Then Sym(#) has a Lie algebra:
Lie Sym(H) = sym(H) := {L: L\H tangent to
Since all our considerations will be local, we can assume that everything takes place in some neighborhood of a fixed point p0 e H; such neighborhood can be lessen finitely many times.
Definition 1.1. The hypersurface H is said to be (locally) affinely homogeneous if:
TP0H = SpanR {L|P0: L e sym(H^.
According to the Lie theory, the 1-parameter groups p 1—> exp(tL)(p) stabilize H, and Sym(H) is then locally transitive in a neighborhood of p0 e H.
The problem of classifying all affinely homogeneous n-dimensional local analytic smooth submanifolds Hn C Rra+C is probably of infinite complexity. Even for n = 2 = c, it is not completed.
In the hypersurface case c = 1 and in dimension n = 2, the classification was completed two decades ago by Doubrov-Komrakov-Rabinovich [2], [3] and by Eeastwood-Ezhov [4], see also [5], [6] for a differential invariants perspective.
In dimension n = 3, and codimension c = 1, all multiply transitive models were classified in [7], while for the a special affine subgroup Saff(R3+1) C Aff(R3+1) was treated and completed in the (unpublished) Ph.D. thesis of Marc Wermann [8], and the works of Eastwood-Ezhov [9, 10].
Jointly with Chen [11], the author has studied a so-called parabolic surfaces H2 C R3, the Hessian of which had a constant rank 1 (see also [6]). Somewhat analogous 5-dimensional CR structures of dimension 5 whose Levi form is of constant rank 1 were studied in [12], [13],
Problem 1.1. Study algebras of differential invariants and classify homogenous models of constant Hessian rank 1 hypersurfaces Hn C Rra+1.
A similar problem can be formulated in the context of CR geometry, cf, [14],
1
hvpersurfaees Hn C Rra+1 in dimensions n = 23 4; these results are presented in a forthcoming paper [14], Exploring then dimensions n = 5, 6, 7, the author was surprised to realize that there are no homogenous models except the degenerate ones obtained by taking a product of Rm with a homogeneous hypersurface Hn-m C Rra-m+1 so that 2 ^ n — m ^ 4,
He then tackled to prove a non-existence result, which, incidentally, provides a complete classification (in the constant Hessian rank 1 branch). But the computational task appeared to be unexpectedly hard, and it took one year to write a detailed proof in general dimension n ^ 5,
The main result of this paper concerns dimensions n ^ 5, but several results are true for all n ^ 2, and will be useful for [15].
In the paper we may emphasize three statements. The first one appears as Theorem 13.1 below.
Theorem 1.1. Let Hn C Rra+1 he a local affinely homogeneous hypersurface having constant Hessian rank 1. Then there exists an integer 1 ^ nn ^ n and affine coordinates (x1,... ,xn) in which:
Hn = HnH x Rn-nH--1 T
is a product of an affinely homogeneous hypersurface HnH C RnH+1 times a ldumb' Rn-nH-1, and is graphed as:
2 2 nH
If 2 /lf,mif if . if ■ \
„ _ I X1X2 I V^ m-1 V^ 1 ____,
= 2+2 + m\ +Xl ^ 2 (i - 1)1(7 - D^0"-1 (3)J
m=3
m„
2 2^ V m\ 1 ^ 2 (i - 1)!(j - 1)!
m = 3 i,j> 2 / w /
i+j = m +1
+ ^ Em(x1,...,XnH),
m=nH+2
with graphing function F = F(x1,..., xnH) independent of xnH+1,... ,xn.
We notice that the variables (x1,..., xnH ,u) are present in such a graphed equation. Of course, we are interested in the hvpersurfaees for which nn = n. Such hvpersurfaces can be called nondegenerate, but we will not use such a terminology. With nn = n, Theorem 1.1 shows the graphing function up to order n + 1 included. Up to order n + 3 included, we prove
Theorem 1.2. In any dimension n ^ 2, every local hypersurface Hn C Rn+1 having constant Hessian rank 1, and which is not affinely equivalent to a product of Rm (1 ^ m ^ n) with a hypersurface Hn-m C Rn-m+1; can he affinely normalized up to order n + 3 as:
2 n m
if** if** if /if1'0 if if if \
1 1 2 1 m m- 1 1
U — ~m\~ + Xl ^ 2 (i- 1)!(7- 1)!)
2 2^ V m! 1 ^ 2 (i - 1)!(j - 1)!
m=3 i,j>2 ^ ' '
i+j=m +1
n+1
I 77 X1 X2 I ^n ST^ 1 Xi X3_
+ Fn+1'10^0 (räTT)! +1 a -1) !(j-1)!
i+j=n+2
™n+3 rrn+2 ™ rrn+2 rrt rrn+2 r
X1 X1 X2 X1 Xi X1 Xn
+ ^n+3,0-0 7——JV7 + ^n+2,10-0 7——+ ^n+2,0010-0 7——+ ' ' ' + ^n+2,0-01 7——7TT (n + 3)! (n + 2)! (n + 2)! (n + 2)!
„n+1
xx X2 X2 ra+1 1 xi Xj
+ ^„„...o-^— + ^ 1 (. - 1} 1{._ 1)(
i+j = n + 3
+ 0X2t...,Xn (3) + 0XUX2 ,...,Xn (n + 4).
Furthermore, linear GL(n + 1, R) self-maps (fixing the 0rigin) = (cd){Xi) of such a hypersurface are necessarily weighted dilations of the form:
1 1 2
yi = CXi, y2 = 0, ys = cx3, ..., Vn = Xn, v = c u, with ce R*.
In other words, the isotropy is at most one-dimensional. The more advanced Theorem 25,1 gives terms of orders n + 4, n + 5, which are more complicated but unfortunately necessary in order to establish our unexpected main result.
Theorem 1.3. In any dimension n ^ 5, there are no affinely homogeneous constant Hessian rank 1 nondegenerate hypersurfaces Hn C Rra+1.
Here is the key reason why homogeneous models do not exist when n ^ 5, In Sections 26 and 27, we shall obtain the two equations I and II shown in Proposition 26,1, that will contradict homogeneity, at the infinitesimal level.
Readers could admit Theorem 1,2 or Theorem 25,1, and go directly to Sections 26 and 27, which are simple to read. Unfortunately, the core normalizations done in the previous sections are hard, require patience, and indurance.
All computations were done on a computer fully in dimensions n = 2, 3, 4, 5, 6, 7, during more than two months of exploration, from December 2020, to February 2021, Especially, all technical statements of this article were constantly checked to be true on a computer in dimensions n = 5, 6, 7, This acted as a guide to set up the general dimension by hand.
In fact, it happened to be unexpectedly hard to write by hand a detailed proof in general dimension, the computer was unable to do that! It was really necessarv to normalize {u = F(x1, ... , xn)} up to order n + 5 as stated in Theorem 25,1, because no contradiction occured in lower order ^ n + 4 for the 'particular' dimensions n = 5, 6, 7,
With slightly harder computations, it can be shown that in any dimension n ^ 2 we always have
dim Lie Sym(#) ^ 4.
Thus, when dim H = n ^ 5, (infinitesimal) homogeneity cannot take place. All statements hold for C instead of R, with the same proofs.
2. Hessian matrix and its rank invariancy
We suppose that coordinates (x, u) are centered at p0 E M, so that p0 is the origin. Denote its image by q0 := ^(p0). We also assume that q0 is the origin in R"+ \ too. Then ^(0) = 0
forces 0 = t1
T0 in (1.1), which becomes a general GL(Rra+1) transformation:
V1 ' " «1,1 • • 0>1,n b1 - x1
yn — 0"n,1 • b1 xn
V . C1 •• • C-n d _ u
(2.1)
Importantly, the basic assumption that ) C K, namely that 0 = — v + G(y) when (y ' °c d) (u) as written above and when u is replaced by F(x), is expressed as the fundamental equation:
0
C1 ^C1 • • • C-rfy ^Cr^ /'' (^C1, . . . , ^Cr^)
+ G (0,1,1X1 +----+ a,1,nxn + 61F (X1,... ,xn),
......, an,1X1 + • • • + + (Xl,..., .
(2.2)
which holds identically in C{x1,..., xn}.
One step further, by composing ^ with two elementary affine transformations, we may assume that the tangent space T0Hn = {u = 0} is horizontal, and that T0Kn = {v = 0} as well. In other words, F = Ox(2^d G = Oy(2).
Thus, order 0 and order 1 terms are absent in the following two expansions:
u = F (x) V = G(y)
E
ï0.....CT,
|-- + CTr
E
CT1>0,...,CTn >0 CT1+-----+CTn>2
I I ^^
ail ■ ■ ■ anl
yl1 ■■■ Vu
G.
ti>0,...,t„>0 ti + -+t„>2
Til ■■■ Tn\
;» (0), (0).
Lemma 2.1. The linear transformation = ("d)(X) sends u = Ox(2) to v = Oy(2) if and only if 0 = c1 = • • • = cn.
Proof We write (2.2) modulo Ox(2):
0 = - Ci x'i-----cn xn - Ox(2) + Ox(2)
□
This is interpreted as a group reduction:
ai,i
0"n,1
Ci
Q>i,n bi
Ln,n bn
d
o>i,i ■ ■ ■ 0,i,n bi
®>n,i 0
1n,n bn 0 d
We then necessarily have
det (a,i,j) = 0 = d.
In what follows, we shall need an equivalent expansion gathering homogeneous terms of fixed order:
« = E w E ■ ■ ■ E ** ■ ■ ■ ^ —(0) =: E pm(x).
m'^2 ii = i im = i m'^2
The proof of this expansion is left for the reader. 2
(2.3)
n n
n n
U
where:
IE EXi2 f™ + O(3) and v = è E E y* yj2 9h,h + °(3)
h=i 32=i
il = i %2 = i
fh,Î2 : = Fx. x. (0) = f%2,ix and 9h,h := yj2 (0) = Qj
32,31'
With x = T(x]^,..., xn) being a column vector, we can abbreviate:
u = |Tx ■ F(2) ■ x + OX(3) and v = iTy ■ G(2) ■ y + Oy(3)
Definition 2.1. At any point p E Hn, the Hessian matrix is well-defined in any system of coordinates (x,u) centered at p = (0, 0), in which, Hn is graphed as u = F(x) with 0 = F(0) = Fx, (0) = ••• = Fxn (0).-
Hessian(F )
Fxixi(0) ■■■ Fxixn(0)
. FXnXl (0) ■■■ FXnXn (0) _
f:
i,i
f:
i,n
L â
n,i
fn
F (2).
At first sight, this definition depends on coordinates, but a key invariancy lies behind.
Lemma 2.2. The rank of the Hessian matrix Hessian(F) is independent of the affine coordinates (x1,..., xn, u) centered at p in which 0 = F(0) = Fxi (0) = • • • = FXn (0).
0
i
•w
C
n
Proof. We take another system of coordinates (y1,... ,yn,v) centered at p = 0 in which the hypersurface is graphed as v = G(y) with 0 = G(0) = Gyi(0) = ■ ■ ■ = GVn(0), namely:
V = Ty ■ G(2) ■ y + Oy(3).
Hence, there exists an invertible linear transformation of form (2,1) with 0 = c1 = ■ ■ ■ = cn by Lemma 2,1, namely, y = A ■ x + B ■ «and v = du, which sends u = F (x) to v = G(y).
Then we write fundamental equation (2,2) modulo Ox(3), observe that Oy(3) = Ox(3) when y = A ■ x + B ■ F(x) = Ox(1) and we factorize:
0 = - u + G(y)
= - dF(x) + G(A ■ x + B ■ F(x))
= - d 1 Tx ■ F(2) ■ x - Ox (3) + |T [A ■ x + B ■ F(x)] ■ G(2) ■ [A ■ x + B ■ F(x)] + Oy (3)
= ^x dF(2) ^A ■ G(2) ■ ^ ■ x + Ox(3),
and deduce, since x G Rn is arbitrary, that:
dF(2) = ■ G(2) ■ A. Finally, since d = 0 = det A, we conclude that rank F(2) = rank G(2),
□
Suppose that H = {u = F (x)} is given in coord mates (x,u) ^^^teed at p0 = 0, with 0 = F(0) = Fxi(0) = ■■■ = FXn(0). At any other point p ~ p0 close to the origin with P = (x1>P,..., xn,p, up), where up = F(xp), we use the centered coordinates:
y1 := X1 - X1-
yn • Xn X
n,p 1
V • U F(Xp) Fxi (Xp) (X1 ■ ■ ■ Fxn (Xp) (Xn Xn,p).
The new graphing function
G(y) •= F(Xp + y) - F(Xp) - Fxi (xp) V1-----Fxn (xp) Уn,
satisfies G(0) = Gyi (0) = ■ ■ ■ terms are of degree ^ 1 in y1,
Gyiyi(0) ■ ■
- Gynyi (0)
Gyn (0), hence Definition 2,1 applies. But since the correction . ,yn, they disappear after the second order differentiation:
Gyiyn (0)
Gynyn (0) J
Fxixi (xp)
. Fxnxi (Xp)
Fxixn (xp)
Fxrixn (xp) -
Lemma 2.3. The Hessian of a graphed hypersurface u = F(x1,...,xn) is a well defined 'matrix-valued function of x:
Fxixi ■ ■ ■ Fxixn Hessian(F)(x) •= : '•. :
. Fxnxi (x) ■ ■ ■ Fxnxn (x) -
whose rank is invariant under affine equivalences (at pairs of points corresponding one to another).
1
We take a hypersurface Hn C Rn+1 with 0 G Hn and T0Hn = {u = 0} graphed as:
1
u = F (x) = Fxx (0) + Ox(3)
=1 =1
Its Hessian at the origin 0 is represented by the n x n matrix (Fxixj (0)),
Lemma 3.1. Whenever (FXiXj (0)) is not the zero matrix, there exists an affine change of coordinates y = Ax, v = u, making nonzero (after renaming y =: x):
Fx,x, (0) = 0.
Proof If there exists an index i* with Fx.itx.it (0) = 0, simply permute affine coordinates to set
^C 1 : ^C ■
Assume therefore that 0 = Fx.x. (0) for all i = 1,..., n. By assumption, there exist i* and j* = i* such that Fxitxjt (0) = 0, We rename x1 := xit to have 0 = Fxixjt (0) and we change j* ^ 2 to be the smallest satisfving Fxix. (0) = 0, We also abbreviate fij := Fx.x.(0), Hence, fu. =0 " > -
Since all diagonal terms are zero, only remains, and we can expand modulo Ox(3):
i<j
U — ^ ^ X^Xj fij ^ ^ X^Xj fij + ^ ^ X^Xj fz,j
i<j i<j i<j j<-j* j*<j
X1Xj. f1,j. + ^ ^ X1Xj f1,j + ^ ^ XiXj fi,j + ^ ^ XiXj fi,j.
j*<j 2^i<j 2^i<j
j*<3
We let xjt := y1 + yjt, while Xj := yj for j = j* and u := v, whence Ox(3) = Oy(3), so that the first monomial becomes
V1 (y1 + yj.) hi.,
with a nonzero coefficient for y1y1. The three remaining sums cannot incorporate y1y1. Thus, the new graph v = G(y) satisfies Gyiyi (0) = 2 f1,jt =0, □
4. Independent and border-dependent jets Up to the end of the article, we shall assume that
Fx.x-, (0) = 0.
Also, our main assumption is that the Hessian matrix (Fx.x (x)) has constant rank 1 for all x ~ 0 in some neighborhood of the origin.
It is elementary to verify that, for constants E R with ip1t1 = 0,
1 = rank
^1,1 <fl,2 • • • <Pl,', <f2,1 V2,2 • • • ^2;,
<P1,1 <P1,3 Pi,1 <fi,j
V 2 ^ i,j ^ n.
<fn,1 fn,2 • • • ^n,n
Main assumption. For all x ~ 0 in some neighborhood of the origin:
FXiXj(x) = FxixiX) ^(X) V ^ = 2,... ,n. (4.1)
"xixi )
By differentiating this identity with respect to x1 ,x2,... ,xn, by induction it is easy to prove that every derivative FxTxi2 ^ (i^th i2 +----+ in ^ 2 and arbitrary t e N is expressed as a
polynomial in the derivatives F ^ i'2 i!n (x^th i[ ^ 1 divided by a certain power (Fxixi (x)) *.
In fact, we shall need to know what formulae hold only for i2 + • • • + in = 2.
Terminology 4.1. • The independent jets are the derivatives with i1 = 0 or i1 = 1:
^x22 ■■■xin ^^ Pxix2 ■■■xin (x) • The border-dependent jets are the derivatives:
FxTxixj(x), t ^ 0, i,j = 2,...,n.
0
The way how these border-dependent jets Fxtx.x. are expressed in terms of the independent jets can be seen by differentiating (4,1) v times with respect to x1. We shall also employ the following abbreviations:
X' := (X2, . . . ,%n), fii,i2,..., in := Fxiixi2 ...X%i (0).
5. Normalization at order 2 Assuming that f1,1 = 0, we begin with the identity
F (x) = fa* xixi + °x(3).
u =
Changing u to -u if needed, we can assume /1,1 > 0. The plain dilation y1 •= \Jf1,1 x1 makes f1,1 = 1, namely, using again the letters x, u:
u = I Xi + X1Xj /1J + 22 XiXj fij + Ox(3).
We should say that our Main assumption is (implicitly) assumed in all statements.
Assertion 5.1. There exists an affine change of coordinates which normalizes u = F(x) to
u = 1 x2 + Ox(3). Proof We first collect all monomials incorporating x1:
2 1 / x 2
U = - I Xi + Xj /1,7 I - 1 I Xi
2<j<n
(xi + ^ xj fuj - 2f ^ xj fijj +1 ^ xixj fij + °x(3)
jXn
V
=: new xi
and we get with a modified fiyj that
u
2 + 2 + °x(3).
2
This implies: Finally, (4,1) yields:
2^i ,j^n
0 = FXlXj (0) for all 2 ^ j ^ n. FxiXj (0) = 0 forall2 ^ x,j ^ n. 12
□
Next, starting from 'U — 2 X1 + Ox(3), the goal is to normalize order 3 terms. But before
2
to group reduction:
GL(n+1,R)
«1,1 «1,2 ■ ■ ■ «1,n bi
«2,1 «2,2 ' ' ' «2, n b2
&n,1 &n,2 ' ' ' @>n,n &n
C1 C2 ■ ■ ■ Cn d
Gstab
«1,1 0,1,2 02,1 «2,2
On,1 On,2 0 0
«1,1 0 «2,1 0*2,2
On,1 On,2 00
01,n
02,n i>2
On,n bn 0 d
G2
s tab
0
6l
02,n
^n,n bn
0 «1,1
0
1
-w
Lemma 5.1. The subgroup G^tab of G1tab which sends 'U — 1 X1 + Ox(3) fo V = 1 y2 + 0(3)
1
stab
consists of matrices in GStab with:
0 = a1,2 = ■ ■ ■ = a1>n and d = a^^. Proof We rewrite (2,2) modulo OX(3) and we equate the eoeffieients at x1x^ ..., x1 xn to zero:
1 2
□
0 = - V + 2 y2 + °y(3)
1 2 = - du + 2 («1,1 £1 + «1,2 ^2 +----+ «1,n Xn + 61 u) + °x(3)
= - d 2 x{ - °x(3) + 2 «1,1 + 0,1,1 X1 («1,2 X2 +-----+ «1,n Xn + °x(2))
+ («1,2 %2 +-----+ «1,n Xn + °x(2)) 2 + °x(3).
6. Normalization at order 3
We proceed to the order 3 terms:
1 ^^ ^ X/0s x^02
u = — X2 + y : j - f01,02,03 + °X(4).
2 t—' >n Ol! ^2! 03!
In the sum, we pick the monomial 1 x\ f\We recall x' = (x2,..., xn). The remaining cubic terms are of the form x2LA(x') + x1B(x') + C(x'). Hence, they are (1), Since they are cubic,
they are products of the form (1) OX(2), Thus,
u = F(x) = 2 x2 + 6 X3, /1,1,1 + °x(1)°x(2) + °x(4),
V = G(y) = 2 y2 + y3 91,1,1 + °y (1)0,(2) + Oy(4), Lemma 6.1. One can normalize g1,1,1 := 0. Proof With free b1 g R, use the map belonging to G2tab:
y1 := X1 + 61 u, y2 := X2, ......, yn := ^n, v := u.
Since y' = x\ hence °y'(1) °y(2) = (1) 0X(2), and fundamental equation (2,2) reads: 1 2
0 = - ^ x2 - 1 /1,1,1 xf - Ox(1) °x(2) - Ox(4)
+ 2 + 612 xf + Ox(3^ +6 91,1,1 {x1 + Ox(2))3 + Ox(1)Ox(2) + Ox(4)
= 6 [ - /1,1,1 + 3 61 + 91,1,1 ] + Ox(1) Ox(2) + Ox(4).
No monomial x\ can appear in remainders. Hence, the coefficient at x\ must vanish. This means that g1,1,1 •= /1,1,1 - 3 ^necessarily, But since b1 is a free parameter in the affine transformation, we can choose b1 •= 3 f1,1,1 to normalize g1,1,1 •= 0, □
To normalize further, we can restart from this v = G(y) having g1}1,1 = 0, call it u = F(x) with f1,1,1 = 0, and again normalize the new target v = G(y) with g1,1}1 = 0, In other words, both hvpersurfaees are normalized similarly (as always):
u = F (x) = 1 x2 + 0 + Ox' (1)Ox(2) + Ox (4),
2
1. .2 2
v = G(y) = - y2 + 0 + °,'(1)0,(2) + Oy(4).
Furthermore, before taking account of the normalizations f1,1}1 that the current stability group G^tab is:
0 and g1.
1,1
0
a,1A 0 ®>2,1 ®>2,2
an,1 0>n,2 00
0
&2,n
b1 b2
Q"n,n bn
0 ah
This being a subgroup of GL(n +1, R), its block-trigonal determinant must be nonzero, whence:
«2,2 • • • a2 ,n
a1,1 = 0 =
a<a,2 • • • an
Next, let cubic terms of the form x2 Ox(1) appear:
1
1
u
F (x) = 2 X21 + 0 + - X22 (<P2 X2 + ••• + Vn Xn) + X1 Ox' (2) +Ox (3) + Ox(4).
2
Lemma 6.2. The remaining cubic terms x1 Ox(2) + Ox (3) = 0 are zero. Proof. It suffices to show that
0 = Fx^xj(0) to all 2 ^ i, j ^ n, 0 = Fx.x.xk (0) to all 2 ^ i, j, k ^ n.
(6.1) (6.2)
By previous normalizations, we have 0 = Fxix.(0) to all i = 2,... ,n. Differentiating (4.1) with respect to x1 and to Xk, we get the following vanishings:
Fxixixi (0) Fx^xi (0) + Fx-^xi (0) Fx1x\xj (0) Fx-^xi (0) Fx-^xi (0) Fx^x^x-l (0)
p (0) _ _=—-—° -° J -°--—°
rx\xixj (0)
Fxixi(0)
Fxixi (0) Fx^xi (0)
0,
Fx
xix jxk
F^xixk (0) Fx-txj (0) + F^xi (0) Fx-txj xk (0) F^xi (0) Fx-txj (0) (0) (0) = - J „°—_ °----„° 1' „ ° _- = 0.
Fxixi(0)
Fxixi (0) Fx1xi (0)
□
Thus, the two hypersurfaces are
u = F(x) = 1 x1 + 0 + 2 x1 (tp2 X2 +-----+ xn) + Ox (4).
V = G(y) = 1 y2 + 0 + 2 V2 2 V2 + ••• + ^n yn) +Oy (4).
Lemma 6.3. The property 0 = p2 = • • • = tpn is equivalent to 0 = ^2 = • • • = Proof. The general map of G2tab which stabilizes the normalization up to order 2 is
y1 = a\t1 x1 + b1 u,
V2 = a2,1 X1 + 02,2 X2 +----+ a,2,n xn + b2 u,
Vn
an,1 X1 + an,2 X2 +-----+ an,n xn + bn u,
a2 1 u.
2
V
Therefore, fundamental equation (2,2) reads as
0 = - a\l
L1 + 2 X1 ^ X2 +-----+ +
+ 2 («1 ,1 Xi + bi1 x! + O.(3))2
+ 1 (ai,i xi + (ajti Xi + a^ x2 +-----+ xn + Ox(2)) + Ox(4).
Of course, the coefficient at xf is zero, Next, picking the coefficients at xf and
at xix2, > > >,
we get:
= - «i,i ^ aj,i, (6.3)
<¿2 = Y, aj,2, ........., <Pn = S aj,n.
Since the (n — 1) x (n — 1) matrix (aj,k) is invertible, we have <p = Ta • hence tp = 0 if and only if ^ = 0. ' □
Before we discuss the two distinct affinely invariant cases p = ^d p = 0, we must examine infinitesimal affine automorphisms.
7. Tangency at order 2 We take a hvpersurface normalized at order 2:
u = 1 xi + O,(3) = F (x). A general affine vector field reads as
L = (ti + Ai,i xi +-----+ Ai,n xn + Bi u^ -j^-
+ (T2 + ^2,i Xi +-----+ A2,n Xn + B2 U^ 9
8X2
+.......................................
+ (Tn + AnA xi +-----+ An,n xn + Bn u]
+ (Vo + Ci Xi +-----+ Cra xn + Du^
is tangent to the hvpersurface {« = F(x)} if and only if:
L{ - u + F(x)) = 0 (7.1)
u:=p (x)
identically in C{xi,..., xn}. By neglecting the terms of order ^ 2, this equation gives:
0 = - To - Ci Xl - C2 x2-----Cn xn - D Oa(2)
+ (Ti + O,(1)) (xi + Oa(2))
+ (T2 + Ox(1)) Ox(2) +..................
+ (Tn + Oa(1)) Oa(2),
whence:
0 = To, Ci = Ti, C2 = 0, ......, Cn = 0.
Hence,
L = (Ti + Ai,i xi +-----+ Ai,n xn + Bi uj
+ (T2 + A2,1 Xi +-----+ Ä2,n xn + B2 uj -
+..............................................................................(7.2)
+ [Tn + An i Xi + • • • + An,nxn + Bnu ) — V / oxn
+ ( Xi +
1 sy TRra— 1
8. Product case Hn = H1 x R We first examine the affinely invariant case ip2 = • • • = ipn = 0:
u = 2 4 + 0 + Ox(4).
Notice then that (6,3) becomes b1 = 0,
Lemma 8.1. If such, a hypersurface is affinely homogeneous, then F(x) = F(x1) is independent of x2,..., xn.
Proof We examine tangency equation (7,1) using L from (7.2) modulo Ox(3): 0 = — T1 X1 — D Q x2 + Ox(4)^
+ (T1 + A1a X1 + A1,2 X2 + ••• + A1,n xn + Ox(2^ (X1 + Ox(3))
+ (T2 + Ox(1^ Ox(3) +..................
+ (rn + Ox(1)) Ox(3). The coefficients at x1, of x1x2, ..., of x1xn must vanish and this gives:
D = 2 Alt1, An = 0, ......, A1,n = 0.
Thus,
L = (T1 + A1A X1 + B1 u)-^
+ (T2 + A2,1 X1 + A2,2 X2 +-----+ A2,n Xn + B2 uj
+ ................................................
+ (Tn + An,1 X1 + An,2 X2 +-----+ An,n xn + Bn u)
V / oxn
+ (T1 X1 +2 Alt1 u)l
Next, let order 4 terms appear:
1 Tai . . . u = -xj + 0 + F4(x) + 0,(5), where F4(x) := V -1-^
/ ±-' (Ti I • • • /T._ I
2 1 v ' ^^ ^ aj •••a.
-----+an =4
Lemma 8.2. This F4(x) = F4(x1) is independent of (x2,... ,xn).
n
W
Proof. We write and examine the tangency equation up to order 3: 0 = - T\ X! - 2 ( 2 xf + Ox(4)
(1 xf + O,(4))
(1 xf + 0,(4)))
+ Ti + X! + Bi x2 + 0,(4) J J (xi + F^ + 0,(4))
+ (T2 + 0.(1)) (F4 +Ox(4)) +...........................
+ (Tn + 0.(1)) (FXn + 0,(4)).
This equation considered modulo 0,(4) is a polynomial in (x1 , X2, ... , ) of degree ^ 3 which must vanish identically:
0 = Tf Ft1 + Bf! x\ + T2 FX2 +...... , ^
+ T F4
1 ±n 1
(8.1)
Notation. [Coefficient picking] For (ti,t2, ... ,rn) G Nra, given a converging power series:
E(x) = E(x1,x2,... ,xn)
£
ry^1 ry 1 2
Xr.
Vf (72 °n !
Ex°1 x°2 •••x°ri
(0),
we denote
[41 x22 ••• C]{E(x)) :
—;-;-t ET-ri T2 t„ (0).
! r2! • • • rn! Xl X2 n
Therefore, in (8.1), the coefficients at ail monomials X1 X 2 • • • X,-^ with T\ + t2 +----+ Tn = 3
must vanish.
Disregarding the monomial xf in order not to let Bf intervene, i.e., taking all t\+t2 + •
3 with T\ = 3, we get:
= h I [Xf Xl2
0= Tl ([
+ ^ ([■
2
1 2
+
+ Tn ([;
2
12
^n ]
№
The following goes almost without saying.
Observation 8.1. [Transitivity] Since T0 = 0, at the origin 0 = (0,0, value of L is
, 0) G Rn+1 the
L
Ti
d
dx1
+ t 9 + +2 -
0 OX2
+ • • • + Tn
d
dxr.
and since also
T0 Hn
Span
d
dx-i
d d
0' dx2 0 dxn
linear relation between T1,T2,... ,Tn can hold only for Hn C Rn+1 being affinely homogeneous.
Consequently, the coefficients of T1, of T2, ..., of Tn above must all vanish. Restricting attention to r1 = 0, and taking all (r2,... ,rn) G Nn-1 wit h t2 + • • • + rn = 3, we obtain:
0 = T1 ([x2? + ^'2 ([:
Xr,-.
4
XI
2
m
x n ] ( F3
+
+ Tn ([42 ••• xn ]« )) ,
0
0
0
0
■n
2
whence:
0
\x
which is equivalent to:
0 = F4 ,
= x
\x
0 = fx4 .
xn
Thus, F4(x) is independent of X2,..., xn, and can be denoted bv E4(x1), □
5
u = 2 x\ + 0 + E4(x1) + F5(x) + Ox(6). We argue by induction. For m ^ 5 we assume:
u = 1 xj + 0 + E4(X1) + • • • + Em-1(X1) + Fm(x) + Ox(m + 1).
To complete the proof of the lemma, we should show that Fm (x) = Em (x1) is independent of
x2, ... ,
Using L from (8,1), we write and examine tangenev equations up to order m — 1:
0
— T1X1 — 2A1A^x21 + E4 + ••• + Em-1 + Ox(m))
+
T1 + A1A X1 + Bl(\x\ + E4 + • • • + Em-2 + Ox(m — 1))
+ E*X1 + ••• + Ex?~1 + Fxn1 +Ox(m)
+ [T2 + Ox(1)] [F™ + Ox(m)] +............................
+ [Tn + Ox(1)] [fx: + Ox(m)].
In the first two lines, all appearing monomials X1 X2 • • • Xfy with n + t2 + • • • + rn ^ m — 1 are such that ^ ^ 1 except the ones in T1 F:.
Therefore, when applying the coefficients-taking operators [xi N"-1 with t2 + • • • + rn = m — 1, it remains only
r-T2 2
](•) for all (T2,..., Tn) E
0
-pm x1
T1 ([x? •••x%](F1 +T2 (K2 ))
+
rrT2
n \
whence:
0
\x
T2
+ Tn ([:
IK )
Xv
*]№m)),
which is equivalent to
0 = Fm,
0 = fx: .
xn
Thus, Fm(x) is independent of x2,..., xn, it can be denoted by Em(x1), and by the induction in m —y to we complete the proof of the proposition:
u
1
~2X
1 + ^Em(x1) =: F(X1).
m=4
□
2
2
2
9. Interlude: expansion of F in homogeneous (in)dependent monomials We expand:
F(X) ^ ] • • • Fai,...,an,
11,...,1„>0
with F1 := 1 Fx< (0), In accordance with Terminology 4,1, we introduce Terminology 9.1. A monomial % 1 x<i • • •
F11,a2,..,an -will be said:
• independent if a2 + • • • + an ^ 1;
• border-dependent if a2 + • • • + an = 2;
• body-dependent if a2 + • • • + an > 3.
Then F decomposes into 3 parts:
F = ^ xaFa + ^ xaFa + ^ x1 Fa.
12+-----h!n<1 12+-----+1n = 2 12+-----+1n >3
If we want to emphasize independent monomials only, the dependent monomials can be gathered as a plain remainder:
F ^ ] X1 X1 • • • xn Fai,a2,...,in + Ox2,...,xn (2).
<1 ,<2
<2 +-----+ an^1
If we want to show also border-dependent monomials, the body-dependent monomials can be gathered as a plain remainder:
f = V^ rfPlrf.12 , , , Tln p + rfPlrf.12 , , , Tln p + O (3)
1 / y ,>J2 -^n 1 11,12,...,1n ~ / J ,>J1 2 ^n 1 11,12,...,1n ' y~Jx2,...,x„\'JJ-
^1,cr2,...,crn>0 &1,cr2,...,crn>0 <2 +-----+ an^1 -----+<n=2
The decomposition of F(x) as a sum of homogeneous terms will be constantly used: F(x) \ 'y ] xi1x22 • • • xn" F(71,(72,...,(7n =: ^ ] F (x).
m=2 11+12+-----+in=m m=2
Then each Fm(x) can be subjected to similar decompositions:
Fm(x) = ^ x1Fa + ^ x1Fa + ^ x1Fc
&1+<?2+-----+(7n=m &i + <?2+-----h(7n=m &i+<?2+-----\an=m
-----V&n ^ 1 -----\an=2 a 2+-----
x° Fi + E
X F u
ai+a2+-----\an=m ai + a2+-----\ari=m
a2+-----han^i &2+-----\an=2
£ x1F1 + Y, x1F1 + Ox,2...„xn (3)
2+-----+ an=m <71 + a 2+-----+ an=-n.
2+-----+ a2+-----+an=2
^ x1F1 + Ox2,...,xn (2).
<1+<2+-----+an=m
a2+-----+<n ^ 1
Finally, we recall the abbreviation:
X : (•^2, . . . , .
3
Disregarding the product case of Section 8, we assume that ( tp2,..., ipn) = (0,..., 0), A natural affine coordinate change
U = + {<¿2X2 +-----+ PnXn) + Ox (4),
2 2 v v y
=: new x2
U = - x\ + - x\ X2 + 0,(4).
enables to normalize:
1 2 1 2 +2
Now we provide a general inductive reasoning. For some integer v with 3 ^ v ^ n, we assume that, modulo dependent monomials which, by Section 9, can be all gathered as a remainder O,' (2), the hvpersurface equation is of the form:
« = y + + ■■■ + f^ + 0,' (2) + +1).
Then let appear all independent monomials of order v + 1: X 2X2 X — X——1
u = T + — + ■■■ + 1^7
if1 if if1 if if1 If If1 If
+ P1 ^ + ■■■ + P-—1 ^f1 + p— ^ + ■■■ + Pn ^ + 0,' (2) + 0x(v + 2). u! u! u! u!
Again, if ( pv,..., pn) = (0,..., 0), a natural affine coordinate change
X i X fX 2 X i X——1
u =t +—+■■■
+ ^J (piXi +-----+ P-—1X-—1 + p—x— +-----+ PnXn) + 0,' (2) + 0x(u + 2),
=: new x
leads to
,-^2 1 r^ if1 if
« = Y + ^ + ■■■ + ^ + + 0,' (2)+ 0,(^ + 2),
and then inductively to
2 2 1 n
U = | + ^ + ■■■ + ^¡T + ■■■ + ^T + 0,' (2) + 0,(n + 2), These are the hvpersurfaees we mainly want to study: they involve all variables x1,x2,... ,xn.
p- = ■ ■ ■ = pn = 0
2 2 —_1
X1 X1X2 X1 X—_1
U = T + + ■■■ + (u — 1)!
+ P1 -yp + ■ ■ ■ + pv—1^— + 0 + ■ ■ ■ + 0 + 0,' (2) + 0,(u + 2).
This degenerate branch will again lead to a product situation.
Since the arguments in the next Section 11 will again involve application of an aifine vector field L to the equation 0 = — u + F(x), and since L is a first-order derivation, we need to know the border-dependent monomials as well. Recall that, by Section 9, body-dependent monomials that are not border-dependent can be gathered as 0,' (3),
To organize properly the thought, we consider simultaneously the two cases 0 ■ and 1 ■ ^-jr by setting up an
Induction Hypothesis 10.1. For some integer v with 3 ^ v ^ n, the hypersurface equation reads as
2 2 v—1 m
if** if** if / if' ' ° if if . if . \ «= |+^+e (^rE 2 (i_x,'_ 1):+flm+1(x1,x2....,xm—1))
m=3 i,j>2 V )• \J )'
i+j = m +1
+ { o } ^^ +x1—1 ^ 1XtXj + R—+1(X1,X2,...,Xv ,...,Xn) + 0x(u + 2), V' i,j>2
where Rm+1 is homogeneous of order m + I in (x1,x2,... ,xm-1), and is of order ^ 3 in (x2,... ,xm-1), where the A^ • = A^ are unknown constants, and where Rv+l is homogeneous of order v + 1 in (x1,x2,... ,xr,..., xn), and is of order ^ 3 in (x2,..., xv,..., xn).
We therefore assume that up to m = v — 1, the values of the border-dependent jets have been found, as they appear within the large parentheses. For v = 3, the formula holds true with empty To complete the induction on u, we need to show that Rv+1 = Rv+1(x1,..., xv-1)
is independent of xr, x^+1, ... , xn5
and we should determine the values of A^. In the nonde-generate branch 1 ■ that we will treat later, we will have to show that A^ = 0 whenever i + 3 = v + 1 and that A^ = (—11—17 when i + j = v + 1,
Abbreviate the homogeneous terms of F (x) normalized up to order ^ u, namely up to m = — 1
U = ^2(x1) + ^3(x1,x2) + -"- + ^^ (x1,x2,.. . ,xv-1) + Ox(u + 1), where, for 3 ^ m ^ v — 1:
ry^^ ry /V» /V»
Nm+1 := ^ +xr-1 E I xx 1M +Rm+1(x1,x2,...,xm-1), m! 2 (% — 1)! — 1)!
i+j=m+1
and abbreviate also the full dependent remainder homogeneous of order v + 1 after { 2 } as:
S ^x1 , x2 , ... , xU , . . . , xn) : xi / J 1 xixj
Ar j + Rr+1 (x1,x2, . . . ,xv, ... , xn) ,
ij>2
which is of order ^ 2 in (x2,... ,xr,..., xn), so that:
u = F (x) = N2 + --- + Nr + { 0 } ^ + Sr+1 + Ox(v + 2).
Lemma 10.1. This function Sv+1 is independent of xv, x~y+1, ... , xn•
Proof. For any two indices v ^ k, I ^ n, we must have by our constant Hessian rank 1 hvpoth-Gsis'
0 — Fxixi (x) ' FXkX£ (x) Fxixk (x) ' Fx:ix:i (x)
— (1 + Ox(1))-(Sxr+x1£ +Ox( !/))
— ({0} (S % + + Ox(u)) ■ ({2} ^ fe + S+t + OxH)
— sxx+x\ + Ox(u) — Ox(2u — 2), since 2 v — 2 ^ v as v ^ 3, and this yields:
o = Sv+1
w iJxk xe.i
Since Su+1 is of order ^ 2 m (x2,..., xn), this completes the proof, □
11. The product case = Hv-1 x Rn"r Here we treat the degenerate branch 0 ■ , We have
u = F (x) = N2(x1) + ■■■ + Nv (x1,...,xu-1) + 0 + Sv+1(x1,...,xv-1) + Ox(p + 2).
Notice that the term Su+1 (homogeneous of order v + 1) depends only on x1,..., xv-1, as does the preceding one Nv.
Lemma 11.1. If such, hypersurface is affinely homogeneous, then F = F(x1,... ,xr-1) is independent of xv, . . . , xn'
Proof. Let homogeneous terms of order v + 2 appear:
u = F = N2 + ••• + Nv + SV+1 + Fv+2(x\,..., xv-\,xv ,...,xn) + O(p + 3).
We claim that Fv+2 is independent of Xp, . . . , Xn*
Indeed, using L from (8,1), we write and examine tangency equations up to order v + 1:
0 =
- Tixi - 2Ai,i (N2 + ••• + Nv + S"+i + Ox(u + 2)j
+
Ti + A1Axi + Bi(N2 + ••• + Nv + Ox(u + 1)) Nxxi + ■ ■ ■ + Nxxi + SX++i + FX+2 + Ox(u + 2)" T2 + A2,ixi + ■ ■ ■ + A2,„Xu + ■ ■ ■ + A2,nxn + B2 (N2 +
Nx32 + ■ ■ ■ + NV + S^ + f;2+2 + Ox(u + 2)"
+ nv-i + OxM)
+
Tp-i + Ap-1Axi + ■ ■ ■ + A^,^ + ■ ■ ■ + Ap-i,nxn + Bp-i (N2 + Ox(3))
Kv-1 + sx+i + fx+I + Ox(/v + 2)'
+
+
Tv + Ox(1) ■ F^2 + Ox(f + 2)
Tn + Ox(i) ■ FT2 + Ox(v + 2)
For all (tv ,..., rn) G Nn-v+i with tv + ■ ■ ■ + rn = //+1, we apply the coefficients-picking operators [x^ ■ ■ ■ xn](•) to this equation,. Since N2, N3, ..,, N^ Sv+i depend only on (xi,..., xv-i), we obtain:
0 = Ti ([x;
i I lx„ ■ ■ ■ x,
](f:
v+2
n xi
+
+ Tv-i ([xv ■ ■ ■ xn ] (Fxv-1
+ Tv ([x? ■■■xn-](f;+2))
+
+ tn ([x- ■■■xn- ](fx+2))
\xTu • • • xTn
X n
](f;„+2)
whence
0 = [x£ which is equivalent to
0 = Fx/+2, ......, 0 = fx:+2.
Xv ' ' xn
Thus, Fv+2 is independent of x v , . . . , x^
and can be denoted Ev+2(xi,... ,xv-i). Next, let homogeneous terms of order v + 3 appear:
u = F = N2 + ■■■ + Nv + Su+i + Ev+2 + Fu+3(xi,...,xv-i,xv ,...,xn) + O(u + 4).
By writing the tangency equation up to order u+2, we realize similarly that Fv+3 is independent of x~v , . . . , xn*
Proceeding by induction, we conclude that
u
F
N2 + ■■■ + Nv + Sv+i + £ Em(xi ,...,xv-i).
m=v+2
□
We now disregard this degenerate ease,
12. Nondegenerate case 1 •
At last, we can start to treat the most interesting branch 1 ■ , Thus, as already stated by the Induction Hypothesis 10,1, we start from
2 2 v — 1 m 1
„ _ x± i x1x2 i V^ I x1 xm i 1 V^ 1 __lPi
U =2+ 2 m! + x ^ 2(i — 1)! (j — 1)! + °x"-1 (3)J
m=3 i,j>2 ^ ' '
i+j=m + 1
+ x^r + xr—1 E 2 xx Avitj + Ox2,...,x— (3) + Ox(f + 2),
where we alreadv know from Lemma 10,1 that Ai • = 0 when i ^ v or j ^ //, and that the body-dependent remainder Ri+1 is independent of
xr, xr+1, . . . , x^
hence is of order ^ 3 in
x2, ... , xv—1.
Here is the statement we mentioned.
Lemma 12.1. We have Aij = 0 whenever i + j = v + 1, and A'ij = (-—1),1(J—1)! wften i + j = v + 1.
Proof. First,
Fxixi = 1 + Ox2,..,xv-1 (1) + Ox(u). Second, for any two indices 2 ^ i,j ^ v — 1:
J-1
^ = T^^r + OX2...,Xv_x (1) + Qx(u),
X1 (i- 1)! J-i
Fxixj = + Ox2,...,x— (1) + Ox(v).
Here, we can also write Ox(1) instead of the more precise (but not useful) Ox2...,xv_1 (1), Third, to compute Fxixj, consider two subcases:
• when 4 ^ i + j ^ v.
Fxx, = x\+'—2 {l—1)^—1)! + x1—1 Aij + Ox(1) + Ox(zv);
• when // + 1 ^ i + j ^ 2 // — 2:
Fxx = 0 + xi—1 A^. + Ox (1) + Ox (v). The vanishing of the Hessian yields: 0 F F F F
— xixi x'x ' xix' xix '
(l + Ox(1) + Ox(u)) ({ 0 } i-^-Di X?3-2 + Ar, + Ox(1) + O») - ((i+ Ox(1) + Ox(u)) ((^ + Ox(1) + Ox(u)),
and it can be expanded into various cases:
• As 4 ^ gives A^. = 0.
• For i + j = v + 1 it gives Ai;i =
• As v + 2 ^ i + j^ 2 i/ — 2, it gives A^- = 0. □ This completes the induction from v— 1 to i/, while 3 ^ v ^ n, cf. Induction Hypothesis 10,1,
13. Summary and beyond We started from arbitrary hypersurface Hn C Rn+1 whose Hessian has constant rank 1 and
2 2 3 4 5
we showed, generally, that one can let appear monomials XX5' ■■■'
the process stops, and we proved the following theorem.
Theorem 13.1. Let Hn C Rn+1 he a local affinely homogeneous hypersurface having constant Hessian rank 1. Then there exists an integer 1 ^ nn ^ n and affine coordinates (x1}. . . , xn), in which
Hn = HnH x Rn-nH--1 T
XnH + 1 i-'-i-^n
is a product of an affinely homogeneous hypersurface HnH C Rnff+l times a 'dumb' Rn-nff-1, and is graphed as:
2 2 nH ( m -, ^
ry'-' ry** ry ry' ' ° ry I ry ry
_ X1 I X1X2 X1 Xm m-1 1 _XiXd____, .
= 2+2 + m! + X 2(i - l)!(j - 1)!+O*2-.^-1 (3)
\ i+j=m +1 J
m=3
+ ^ Em(x1,...,XnH),
m=nn+2
with graphing function F = F(x1,..., xnH) independent of xnH+1,... ,xn. □
Disregarding the product cases (branches) nH = 1,... ,nH = n — 1 which lead to similar considerations in lower dimensions, we will from now on study the class of n on degenerate hvpersurfaees, those involving all variables (x1}... ,xn):
2 2 n m
ry** ry ** ry / ry'' ° ry I ry ry \
«= f + xf + E ^+xT1 E 2 (i—— 1)! + O-2.....(3) + O-(n+2).
m=3 2 ^ ' '
i+j=m +1
The next (substantial) task is to examine further the remainder O x(n + 2).
We gather all remainders OX2)...)Xm_1 (3) as Ox(3), where x' := (x2,... ,xn), and we let appear
the independent homogeneous monomials of order n + 2, namely,
xn+1xxn+1xn xn+1x
x1 x1 x1 x2 x1 xn
Fn+2,0,...,0 7—, „m + Fn+1,1,...,0 7—TTTT + ' ' ' + Fn+1,0,...,1
(n + 2)! (n + 1)! 1 1 (n +i)\-
A reasoning similar to that in the proof of Proposition 12.1 shows that border-dependent monomials of homogeneous order n + 2 have the same form, hence the equation of the hypersurface reads as
2 2 n m i u = X± + XiX2 + ^ fX1Xm +Xm-1 ^ 1
i,3> 2 + j=m +1
m-1 1 _XiXj_\
1 ^ 2 (i — 1)!(7-1)!/
2 2 m! 1 2 ( - i)!( - i)!
m=3 2 ^ ' '
Xn+1XXn+1Xn Xn+1X X1 X1 X1 X2 X1 Xn
+ Fn+2,0,..,0 7-TTÂT + Fn+1,1,...,0 7—TTVi" + ' ' ' + Fn+1,0,...,1
(n + 2)! (n + 1)! ' -'+1>0>...>1 (n +1)!
+Xn E 2 u-XX'-1)! + O"(3) + °^(n+3).
(* - W - 1)!
i+j=n+2
Question 13.1. How to normalize these order n + 2 independent coefficients Fn+2,0,...,0,
Fn+1,1,...,0> ■ ■ ■ , Fn+1,0,...,1 ?
n + 2
( 1 , . . . , n, )
ViVj
V =
2 2 n y± + Élâ + v
m=3
'Vl Vrn m-1 + U 1
m!
E 2
+ G.
n+2,0,...,0
yi+1yi
+ G
(n + 2)
+y- E 1 ViVj
n+1,1,...,0
i+j=m+1
yni+1y2
(i - 1)!0" - 1)
ï)
+ ■ ■ ■ + G,
i+j=n+2
(i - 1)!(j - 1)!
(n + 1)! + Oy' (3) + Oy (n + 3).
n+1,0,...,1
yni+1 yn
(n + 1)!
2
n + 1
^stab
Gn+i
stab
a1,1 0 ■■ ■ 0 61 1 2 " ? 0 ■■ ■ 0 ?1
a2,1 a2,2 ■ ■ a2,n 2 ? ? ■ ■ ? ?
an,1 ara,2 ' ■ an,n bn ? ? ■ ■ ? ?
0 0 ■■ ■ 0 ah . 0 0 ■■ ■ 0 nn+1 a1,1
Step 2. Determine how this subgroup acts on order n+2 terms, and normalize those coefficients among Gn+2.1.....o, Gn+1.1.....0^ ., Gn+1.o.....1 which can be normalized.
This computational task being nontrivial, let us start in low dimensions. The calculations of the next Section will not be detailed, and the remainders Ox(*) will not be written.
15. Stabilizing order n + 1 terms in dimensions n = 2,3,4,5,6 n = 2
2 2 3 3
O^ O^^ J"' J" , J"' J"
.1 X±X2 „ X1X1 X±X2 U —--1---h T4 0--T r3 1 -,
2 2 4,0 24 3,1 6 '
the stability group is
G3
Gstab •
a±± 0 -a\±a2,i a2,i 1 b2 0 0 a21,1
and its action gives:
40 1
- 24
0 = - 24 a2± ^4,0 + 24 a'1,1 G4,0 + - a2,1 G3,1 + - al,± a2± + 7 a
0 31 1 ~2 V , 1 .3
6
8
4
1,1
- r an Fs,1 + — an G3,1. 66
The free group parameter 62 can be used to normalize G4.0 := 0, • n = 3
2 2 3 2 2
,-yi O^^ ' ^-yi^'^yi^'
x 1 t/y^jy'2
U = T + ^T + +
4 4 4 3 x1 x1 x1 x2 x1 x3 x1 x2 x3 + F5.0.0 ^ + F4.1.^ --H F4.0.1 ---+
120
24
24
2
2
3
2
the stability group is
r4
rstab ■
«3,1
0 0
1 0
0 1
«1,1
0 0
-0*1,10*2,1
102 _ 2
' 202,1 301,1 2 3
and its action gives 500 1
0 - - 120 «M F5,0,0 + 120 a0l,1 G5,0,0 + 214 <0^,1 a2,1 G4,1,0 + 112 O^ 02,1 03,1
1
1 a3,1 r4,0,1 + 77: a1 1 03
+ 24 01,1 0i,1 G4,0,1 + a\,1 63 410 1 o „ 1
12
0 — - 24 01,1F4,1,0 + ^ o4,lG4,1,0, 401 1 1 1
02,1
0 — — 24 aM F4,o,1 + 24 ^3,1 G4,0,1 +12 a2,1
The free group parameter a2,1 can be used to normalize G4,0,1 := 0, and the free group parameter b3 can be used to normalize G5 0 0 := 0,
• In dimension n — 4:
X
M = f +
23 X1 X2 X1 X3
the stability group is:
2
+
6
+
22 12
2
+
4
X1 X4
23
+
X1 X2 X1 X2 X3
2
+
555 X1 X1 X1 X2 X1 X3
+ F6,0,0,0 + F5,1,0,^n + F5,0,1,0
+ F5,0,0,1
720
5
ry' rf>
14 120
120
120
24
+
4
2
42
12
32 + X31 X22 X3 +
X1 X2 X4 X1 X3
6
+
8
r5 . rstab .
and its action gives 6000 1
01,1 0 0
02,1 1 0
03,1 0 1 «1,1
4,1 0 -2-2,1 «1,1
0 0 0
0 0
0
1
«1,1
- 01,1 02,1 - 202,1 - \01,103,1
201,104,1 - 02,103,1 b4
1,1
1
0 00 - 720 02,1 F6,0,0,0 + 720 01,1 G6,0,0,0 + 72 0M 02,1 + 48 0M 02,1 04,1
1
120
+ 717 G5,1,0,0 0011 02,1 + 12Ô G5,0,1,0 03,1 + 71^ G5,0,0,1 04,1 + 71 01,1 b4
1
120
1
48
0 51-0 1 02 F + 1 05 r
0 — - 120 01,1F5,1,0,0 + 120 01,1 r5,1,0,0,
0 50-0 1 02 F + 1 04 r 1 02 02 1 03 0 + 1 03
0 — - 120 ,1 F5,0,1,0 +120 01,1r5,0,1,0- 24 01,102,1- 60 01,102,1 + 36 01,1
03,1
5001 1 2 1 3
1
2
2,1
0 00 — 120 a2,1 F5,0,0,1 + 120 a1,1 G5,0,0,1 + 24 a1,1
The free group parameters a2t1, a3t1, b4 can be used to normalize G5i0i0i1 := 0 G5t0t1i0 G6,o,o,o := 0,
4
2
5
2
0
• In dimension n = 5:
" = i +
2 3
+
22 12
+
5
ryry 120
+
2 4
^y»^ ry^ 12
+
32
I I
+
+
2
4
^ 1X4
4 4 2
ry^ ry ^ ry ry^ry**
2 3 3
ry ^ ry ' J ry ' J ry _ ry .
6
+
8
ry6^^ ry6 ry ry6 ry ry6^^ ry6 ry
X1X1 X1X3 XiX4 X1X5
+ ^7,0,0,0,0 _„ . „ + ^6,1,0,0,0 + ^6,0,1,0,0 + ^6,0,0,1,0 + -^6,0,0,0,1
5040
720
720
720 1
1 5 5 5 1
25 33 4 2 42 5 5
+ t: x
2 12 3
the stability group is:
8
12
12
24
720
G6
stab •
«1,1 0
«2,1 1
«3,1 0
«4,1 0
«5,1 0
0 0
5«S
0 0 0
0 0 0
1 «1,1 2«2,1 0 1 0 0
<1 «1,1
_ 10 "3,1 3 «1,1 5«2,1 n6 1 tt1,1
0 0 0
— 01,102,1
1 2 2 — 2 ß2,1 — 3 «1,1^3,1
— «2,1«3,1 — 2 0,1,1«4,1
2 2 2 «2,1«4,1 — 3 Q3,1 — 5 «1,1«5,1
1,1
and its action gives:
0
70000
1
«21
1
1
1
7,0,0,0,0
— 7 ^ 5 5
+ 5040 a1,1 ^7,0,0,0,0 +144 «1,1 «3,1 «4,1 + 240 a1,1 a2,1 «5,1
5040
1 r1 ^6 ^ I 1 r^ „6 ^ I 1 /-1 „6
tTßinnnf/.
720 1
+ ^6,1,0,0,0 «6 1 «2,1 + ^TTT ^6,0,1,0,0 «6 1 «3,1 + ^777 ^6,0,0,1,0 «6 1 «4,1
720 1
720
+ 720 G6,0,0,0,1 «1,1 «5,1 + 240 al,1
0 61000 1 „2 ci , 1 „6
— 720 Q1,1 F6,1,0,0,0 + 720 a1,1 ^6,1,0,0,0)
60100
1
720
— «2,1 ^6,0,1,0,0 + 710 a1,1 ^6,0,1,0,0 — — «1,1 «2,1 «3,1 + 1 «1,1 «2,1 + 1 «1,1 «2,1 ^6,0,0,0,1
1
36
1
48
1
144
1
2T6
— a4 1 «3,1 ^6,0,0,0,1
1 4 1 4 -ßi 1 O2 1 ^6 0 0 10 + - «1 1
360 11 2,1 6,0,0,1,0 144 1,1
«4,1
0 60=10 1 ^2 F + 1 „4 r 1 2 „2 1 3 n r + 1 3
0 = — 720 ,1 ^6,0,0,1,0 + 720 a1,1 ^6,0,0,1,0— 32 a1,1 a2,1 — 144 a1,1 a2,1 ^6,0,0,0,1 + 72 a1,1
0 60=01 1 ^2 F + 1 ^3 r + 1
0 = — 720 a1,1 ^6,0,0,0,1 + 720 a1,1 ^6,0,0,0,1 + 80 a1,1
«3,1
«2,1
The free group parameters a2,1; a3,1; a4,1; 64 can be used to normalize G6,0,0,0,1 := 0 G6,0,0,1,0
0 ^6,0,1,0,0 := 0 ^7,0,0,0,0 :=
• In dimension n = 6:
22 3 224 233
•A, 1 x^x2 x^x3 x ^2 x 1x4 x 1X2 x 1x2x3
u = T + ~1T + ~fT + ~T~ +14 + + 2
5
ryry
X1X5 120 16 +120 +
24 12
442
ry^t ry ry ry^* ry**
+ + +
6
8
25
ry*' ry'-' 12
5 33 5 4 2 5 42 1 5 1 5
2 +3 X1X3X3 + 8 xlx2x2 + 12 xlx2x4 + 12 X1X3X4 + 24 X1X2x5
7 7 7 7
ry ' ry ry ' ry ry ' ry ry ' ry
X1X1 X1X2 X1X3 X1X4
+ ^8,0,0,0,0,0 + ^7,1,0,0,0,0 rr. , „ + ^7,0,1,0,0,0 rri , „ + ^7,0,0,1,0,0
40320
5040
5040
5040
2
6
2
6
b
5
3
a
1,1
2
0
1
1
5
15
7 7
_ X-X5 j-i X-X6 I 53 i 6 2 5 3 4 15 4 2 2
+ ^7,0,0,0,1,0 5040 + ^7,0,0,0,0,1 5040 + 8 X5X3 + ^ X-X4 + 2 ^^ + X1X2X2
5 4 3 1 5 2 1 6 1 5 1 2 6 1 6
~H X1X2X4 ~H 7-7 X5X2X5 X1X3X5 ~H 77 X5X2X3X4 ~H ' — X-X'2 + 1 on x'ix'2x'Q,
6 124 8 the stability group is:
48
2
2
120
a-,- 0
a.2,1 1
a3,1 0
a4,1 0
a5,1 0
5a2,i
^1,1
00
and its action gives:
0 0 0 0
0 0 0 0
1 ai,i 0 0 0
2a2,i a1,l 1 a1,1 0 0
_ 10 «M 5a2,1 1 0
3 a1,l a3 - 15 ^ — al,l a 3 a1,1 2 45 a2,1 IA«3,1 2 a4,1 10 a?,1 a3 a1,1
r a4,1 5 a2 a ,1 —9 aa4A a41,1 1 a1,1
0 0 0 0
— a1,1a2,1 — 2a2,i — §ai,ia3,i
-a2,ia3,i — 2ai,ia4,i
2 2 2 a2,1a4,1 — 3a3,1 — 5a1,1a5,1
15 a2,ia5,i — 3a-,ia6,i — 3a3,-a4,i
1,1
0
800000
1
+
40320 1
a1 1 ^8,0,0,0,0,0 +
1
40320 1
a7 1 G8,0,0,0,0+
7 1 7 1 5040 G7,1,0,0,0,0 aM ^ + 5040 G7,0,1,0,0,0 aM ^ + 5040
Gr,0,0,1,0,0 al 1 a4,1
1 7 1 7
+ rnA n G7,0,0,0,1,0 a1 1 a5,1 + rnA n G7,0,0,0,0,1 al 1 a6,1
5040 5040
1 6 2 1 + -, -, ^^ a- - a4 - +
71000
1152 1
6 1 6 1 1440 aM ^ aM + 720 aM ^ a5,- + 1440
a61,1
1
0- 1 G7,1,0,0,0,0,
al 1 F7,1,0,0,0,0 + _„ .„ a1 1
701000
5040 1,1 7,1,0,0,0,0 5040 5040 ai,i F7,0,1,0,0,0 + 5040 aQ,i G7,0,1,0,0,0— 444 at,i a2,i a4,i + 48 °i-i a2,i a3,i
1 2 4 1 4 2 1 96 aM a2,1 216 ai,ia3,1 2520
a1 1 a2,1 G7,0,0,1,0,0
+ 7717 a4 1 a2,1 a3,1 G7,0,0,0,0,1 — 7717 a3l1a21 G7,0,0,0,0,1
252 336
1 a4 n G 1 -4 „2
a1 1 a4A G7,0,0,0,0,1 —
1008 1,1 4,1 7,0,0,0,0,1 1008
1 5 s~i 1 5
1512 ai- a3,1 G7,0,0,0,1,0 + 720 a1,1
a41,1 a22,1 G7,0,0,0,1,0
a5,1
700100
1
a\ 1 F7,0,0,1,0,0
+ 5040 al,1 G7,0,0,1,0,0 + 224 a3,1 a2,1 G7,0,0,0,0,1
5040
1 4 s~i 1 2 3 1 3
504 ai-i a3,i G7,0,0,0,0,1 + 48 ai- a2,i — 48 ai- a2,i a3,i
1M a4,1 a2,1 ^M,0,-0 + 288 a4,-
a4,1
700010
1
1
a\l F7,0,0,0,1,0 + rn/in a1,1
^lA G7,0,0,0,1,0
5040 m - 1,0,0,0,1,0 ' 5040 560 a3,i a2,i G7,0,0,0,0,i— 80 ai,i a2,i + 240 a1,i
a3,1
700001
1
5040
a2 1 F-,
1
1
1 ^7,0,0,0,0,
1 a3 1 G7,0,0,0,0,1 + 777777
5040
360
1,1
2,1
3
a
1,1
6
3
2
6
0
0
0
0
2
0
The free group parameters a2,i, «3,1, «4,1, «5,1, 64 can be used to normalize GV,o,o,o,o,i := 0,
^7,0,0,0,1,0 := 0 ^7,0,0,1,0,0 := 0 ^7,0,1,0,0,0 := 0 ^8,0,0,0,0,0 := 0-
Instead of attempting to dominate the combinatorics of such formulas in any dimension n ^ 2, we shall inûnitesimalize the determination of the stability group at order n + 1, and also, we shall inûnitesimalize its action on coefficients of order n + 2,
16. Tangency at order 2 in dimension n
Before starting, in any dimension n ^ 2 and in continuation with Section 7, let us examine the tangency of L up to order 2 to the hvpersurface U> — 1 X1 + Ox(3). Thus, in (7.2), we let
T1 := 0,
T
J- n
0
L = ^1,1 X1 +-----+ ^1,ra xra + B1 uj
+ (^2,1 +-----+ M,n ^n + ^2 uj
+
+ (^n,1 +-----+ ^n + uj
+
d d au
Lemma 16.1. Tangency ^2(L(-m+f) | f) up to order 2 holds if and only if the coefficients matrix of L reads:
¿1,1 0
^2,1 ^2,2
00
12
0
A
0 2A11
Proof We write the graph as m = 1 xj + Ox (3), and compute modulo Ox(3): 0 = - D (2 x? + Ox(3))
+ (^1,1 X1 + ^1,2X2 + ■ ■ ■ + ^1,ra+ B1 Ox(2)) (x1 + Ox(2))
+ (^2,1 + ^2,2 X2 + ■ ■ ■ + ^2,ra xra + ^2 Oœ(2)) (0 + Oœ(2)) +......................................................
+ (a„,1 X1 + Ara,2 X2 + ••• + Ara,ra xra + 5ra Oœ(2)) (0 + Oœ(2)). The coefficients of of ..., of x1xra must vanish, which concludes:
D = 2 ^1,1, ^1,2 = 0, ......, ^1,ra = 0.
□
17. Tangency in dimensions n = 2,3,4,5,6
Before proceeding to treating the crucial order n + 1 in general dimension n ^ 2, let us show what formulae exist in low dimensions.
2
• In dimension n = 2, with
u
2 2 X i X iX2
T + :
L = (AM Xi + Ai,2 X2 + Bi u) dXl + (A21 Xi + A22 X2 + B2 u) dX2 + the tangency equation in orders not exceeding 3
u) du
0 — - Cixi -C2X2 — D (X + ^
+ ( AiiXi + Ai,2X2 + Bi -2
f) (xi + XiX2)
gives at order 1 as we know 2
+ (A2,iXi +A2,2X2) ,
Ci := 0, C2 := 0, D := 2Ai,i, Ai,2 := 0,
and it remains 0
2 Ai,i (X2 + XiX2) + (a1a Xi + Bi X^J (xi + XiX2) + (A2,i Xi + A2,2 X2)
2 i
— Xi
\Bi + 1 A2^
+ XiX2
— Ai,i + Ai,i + ^2,2
3
Bi := - A
2,i,
A
2,2
L
Ai,i Ai,2 Bi A2 ,i A2,2 B2 Ci C2 D
Ai,i Ai,2 Bi A2 ,i A2,2 B2 0 0 D
Ai,i 0 Bi A2 ,i A2,2 B2 0 0 2 Ai,i
Ai,i 0 A2,i 0 00
-A2ti B2 D
n = 3
22 3 22
ry*** I" ' I" ,-yi^'^yi^'
x i x 3
u =T +— +—,
L = (.Ai,i Xi + Ai,2 X2 + Ai}3 X3 + Bi u) + (A2,i Xi + A22 X2 + A2,3 X3 + B2 u + {A3,i Xi + A3,2 X2 + A3,3 X3 + B3 u) dX3 + (Ci Xi + C2X2 + C3X3 + D u) d,, the tangency equation up to order 4, the expansion of which can be done, is
X2
2 2 3
— Ci Xi — C2X2 — C3X3 — D + XiX2 + ^J
+ yAi>i Xi + Ai,2X2 + Ai,3X3 + Bi
+ ^A2,i Xi + A2,2X2 + A2,3X3 + B2
X3
+ (A3,iXi + A3,2X2 + A3ßX^ -6-.
Xi . ^1^2 T + ~
X X
Xi + XiX + -2- j
X
i 2
— + XiX2
)
2
0
0
i
2
3
i
2
Starting from order 2 thanks to Lemma 16,1, the matrix reductions read
2
^1,1 0 0 #1
^2,1 ^2,2 ^2,3
,1 ^3 ,2 ^3 ,3
0 0 0 2^1,1
^1,1 0 0 -^2,1
^2,1 0 0
^3,1 ^3,2 ^3,3
0 0 0 2^1,1
^1,1 0 0 -^2,1
^2,1 0 0 2 A - 3^3,1
^3,1 0- -^1,1
0 0 0 2^1,1
In dimension n = 4, with
X i
u
2 +
2 3
ry ry ry ' J ry
+
X2 X2
1^2
4
+ ^1^4 +
„2^,3
ry*' ry'-' ryKJ ry ry
1 2 1 2 3
+
2 6 2 24 2 L = (Ai,i Xi + Ai,2^2 + ^1,3X3 + ^1,4X4 + ßi ti)<9:
+ (^2,1^1
+
+ (^4,1^1 + ^4,2^2 + ^4,3^3 + ^4,4^4 + +
2
£3
£4
one can confirm that up to order 4, the same equations appear as in dimension n =3, and hence we can replace the coefficients of L from what has been obtained just above, so that the tangenev equation up to order 5 becomes
2
0 =
2 2 3
_ 2 A11 ( ^ + ^ + ^ +
1,1 V 2 2 6 24 J
X1X4 \
+ ^1,1 X1 - ^2,1
2
2
23 ^2 ^1^3
+ ( ^2,1 - 3 ^3,1
2
22 ry*' ry*' ry
1
~2 +
6
X1 ^4
2
XTX3
£1 + X1X2 +--— + „ ,
26 23
f + 4*2 +
+ ^3 ,1^1 - ^1,1^3 + B3
x
X1
e +
+ (^4,1 £1 + ^4,2 X2 + ^4,3 £3 + ^4,4 £4) ( ^
5
^1,1 0 0 0 -1^2,1 " 4 " ^1,1 0 0 0 - 2^2,1
^2,1 0 0 0 -1^3,1 ^2,1 0 0 0 - 3^3,1
, 0 -^1,1 0 ^3,1 0 -^1,1 0 -1^4,1
^4,1 ^4,2 ^4,3 ^4,4 ^4,1 0 -2^2,1 -2^1,1
0 0 0 0 2^1,1 0 0 0 0 2^1,1
• In dimension n = 5, with:
X i
M = f +
2
+
3
ry'-' ry _
X1X3
+
2 2 4 2 3 3
ry*' ry*' ry-* ry ry*' ry'-' ry'-' ry ry
2
24
5
ry'-' ry
Xj X5 120
24
ry*' ry^ 12
4
3 2 ^1X2^4
+ X1^2X3 + ---+
2
42
ry^t ry*' 13
2
4
1
2
5
6
2
8
the tangency equation up to order 5 is
, x2
0
ry2 ry2 ry ry3 ry ry4 ry ry5 ry
2 A I + x1x2 + xix3 + xix4 + xix5
- 1,1 {Y "120"
+
c
+ Ai,i xi - A
2,1
2
kL> 1 i +
2
x x2
+
3
x x3
+
4
x x4
~2T
x + x x + x2x3 + x-x4 + X1 +xix2 + — + +
A2,ixi - 3 A31
2
X 1 ^ +
2
23
ry2 ry rf>KJ rf>
2
+ ( A3,ixi - Ai,ix3 - ^A4,1
x 2 +
6 2
x"x2
x 1 2 "2" + xlx2 + 2
3
x x3
x4 x4
x f +
3
x3 x2
+
xjx-3\
4 J
+ A41 xi - 2 A21 x3 - 2 A,1 x4 + B4
2 Jb 1
T
44
xi xix2 \ 24 + "6"/
+ {A5,1 xi + A5,2 x-2 + A5,3 x3 + A54 x-4 + A5,5 x5) 1x0, and the matrix reduction is
5
Ai,i 0 0 0 0 - 2A2,i
A2,i 0 0 0 0 3-^,1
A-,i 0 -Ai,i 0 0 - 4A4,1
A4,1 0 -2A21 -2Ai,i 0 B4
A5,1 A5,2 A5,3 A5,4 A5,5 B5
0 0 0 0 0 2Ai,i
Ai,i 0 0 0 0 -\A2,i
A2,1 0 0 0 0 --As,i
A-,i 0 - Ai,i 0 0 -
A4,i 0 -2 A2,i -2Ai,i 0 2 A - 5 A5,1
A5,i 0 10 A - ya-,i -5A2,i -3 Ai,i B5
0 0 0 0 0 2 Ai,i
In dimension n = 6, with:
2 2 3 2 2 4 2 3 3 5 2 4 4
ry*** ry**ry ' ry**ry** 1 ry*** ry'-1 ry'-'ry ry ry1-" ry )" )" I" I" ry
•AJ i JjiJJ2 JJIJJ2 JJIJJ4 JJIJJ2 .AJI.AJ2.AJ3 JJIJJ5 JJIJJ2 3 2 JJIJJ2JJ4
U = T + ~ + T + ~ + 14" + — + + T205 + ~ + x3x2x3 +
ry*4 ry2 ry6 ry ry2 ry5
+ xx- + x^ + ^ + 5x3x2x3 +
5
8 720 ' 2 '3
6
x2
0 = - 2 A11
xix2x3
12
xix2x4
ry2 r*r'2 ^ ry3 ry ry4 ry ry5 ry ry6 ry S\
1 <-0 i^y 2 Jb iJb 4 \
T + ~1T + ~6T + + T2O + ^20")
{2
(
+ Ai,ixi - A2,1
x
2
f +
2
x"x2
+
3
x"x3
+
4
xix 4
+
5
ry'-' ry
Jb 1 Jb 5
T20"
/ ry2 ry ry3 ry ry4 ^y ry5 ry \
I , , xix3 . xix4 . xix5 . xix6 \
(vxi + xix2 + ~ + ~6T + 14" + T2cT )
+
t
2
A2,ixi - 3 A3,1
2
2
„3,
4
xi xix2 xix3 x "x 4
T + ^T + + ~24~
1 5 1 5
1^x1x3x4 + 24x1x2x5
ry3 ry ry4 ry ry5 ry \
ds id-'3 KL/I^J 4 U/5 \
x 1 2 .3 .4
-1 + x^x2 + + +
2
6
24
5
6
+ Us,1^1 - ¿1,1^3 - 4 ¿4,1
2 2 3
++ ^1^2 ++ X1X3
2
6
,-y»3 ry3 ry ry4 ry ry5 ry S\.
1 Jy^Jy 3 JL> 1JL> 4 \
~6~ + IT + ~ + 12~ y
22 T +
+ ( ¿4,1 £ - 2 ¿2,1 £3 - 2 ¿1,1 £4 - 5 ¿5,1 + ( ¿5,1 £ - ^3,1 £ - 5 ¿2,1 £4 - 3 ¿1,1 £5 + ^5
4
24 +
4
"6"
+
12 )
2 1
T
+ (^6,1 #1 + ^6,2 x2 + ^6,3 x3 + ^6,4 + ^6,5 + ^6,6 Tl^D
and the matrix reduction is
55
ry'-' ry'-'ry \
1 1 2 \
121) + ~24~ y
£6
¿1 ,1 0 0 0 0 0 -1^2,1
¿2,1 0 0 0 0 0 -1^3,1
3, 0 -¿1,1 0 0 0 -1^4,1
¿4,1 0 -2^2,1 -2^1,1 0 0 - §¿5,1
¿5,1 0 10 A - ,1 -5^2,1 -3^1,1 0
¿6,1 ¿6,2 ¿6,3 ¿6,4 ¿6,5 ¿6,6
0 0 0 0 0 0 2^1,1
¿1,1 0 0 0 0 0 - ^2,1
¿2,1 0 0 0 0 0 - 3^3,1
, 0 -¿1,1 0 0 0 2 A
¿4,1 0 -2^2,1 -2^1,1 0 0 2 A - 5 ¿5,1
¿5,1 0 10 A -5^2,1 -3^1,1 0 2 A - 6^6,1
¿6,1 0 -5^4,1 -10^3,1 -9^2,1 -4 ¿1,1
0 0 0 0 0 0 2 ¿1,1
18. Projections and ^^ntd(•) For each integer m ^ 1 and each power series vanishing at the origin
E (x) = E (xi,...,xra) = ^ x?1 • • • E^,...,^,
cti+-----h<Jn^1
we define:
^ (E(x)) := £ ^ •••xs"E01,...,ffB.
ct1+-----h<rn^m,
Several times later, the following elementary fact will be employed.
Observation 18.1. Given two integers g N and two power series H1(x) g O^fc1)
and H2(x) g O;u(fc2), namely,
= ^Vi,..,^!1 , #2(2:)
11 H-----hiJn^fcl 11 H-----+0"n^fc2
/or eacft (homogeneous) order m ^ 0, we have
m— fci
^ ] ^2,<ri,...,<r„ £1 ' ' ' ,
ifte right-hand side being understood as 0 wften m < min (fci, fc2)
6
7
Since our hypersurfaces Hn C Rra+1 have constant Hessian rank 1, independent monomials are of interest. Accordingly, we define
'ind (E(x)) := ^ (^Ei+i,o,...,o x\+1 + Ei,i,.„o x"x2 +-----+ Ei,o,...,i x\xn^j ,
i=0
and also
'ind := ' ° 'ind = 'ind 0 ' ,
that is
(E(x)) := ^ (Ei+i,o,...,o x\+1 + EiA,...,o x\x2 + ■ ■ ■ + Eifl...,,i x\xn^j.
Observation 18.2. Given two integers k1, k2 G N and two power series H"(x) G Ox(k1) and H2(x) G Ox(k2), for each, (homogeneous) order m ^ 0, we have
<nd (Hi ■ H2) = (<ifc2 (Hi) ■ K?n-k1 (H2)),
the right-hand side being understood as 0 when m < min (k1, k2).
We shall need a notation to select homogeneous monomials of order exactly equal to some fixed integer m ^ 1 (notice that the index m is lower-case now):
'm (E(x)) := x\1 ■ ■ ■ xn" Ea1,...,an , ctiH-----+a„=m
and also to select the independent homogeneous ones:
'nt (E (x)) := Em,o,...,oxT + Em-1,1,..,oxi "x2 + ■■■ + Em-1,o,..,1 "xn.
n + 1 n
n - 1
(x",. .., xn-1, u), we know that the hypersurface equation up to order n - 1 + 1 reads as
2 2 n—2 m
ry2 ry2 ry / ry''° ry I ry ry
->J1X2 X > I jy" -A,m m— 1 X > -L JytJyj
2i + ^T + ^V m! +x" ^ 2 (i - 1)1(7 - 1)!
m=3 ¿,j>2
i+j=m + 1
^C ^ ^C/fy_1 ^_2 T 1 xlxj ^ . . ^ . .
+ (n - 1)! + x 2 (i - 1)!(j - 1)! + 0^2,...,^"-1 (2) + 0^i,^2,...,^n-1 (n +1),
i+j=n
with explicit independent and border-dependent monomials, A linear (vanishing at the origin) aifine vector field reads as
d d d
L = Xi — + ■ ■ ■ + Xra_i --+ U —,
ox-" oxn_1 ou
with:
Xi = Ai,ixi +-----+Ai,n_ixn_i + B-u,
Xn_1 = An_1,1 x1 + ■ ■ ■ + An_1,n_1 xn_1 + Bn_1 u,
U = C-x- +----+ Cn_i xn_i + D u.
We consider tangenev of L to such a hypersurface u = F(x",... ,xn_1) up to order n - 1, namely, the condition
0= (L (-u + F)U),
which is equivalent to
0 = nl
,n— 1 ind
(L - + F )U ).
We start arguing from Lemma 16,1 at order 2 considered in dimension n — 1.
Induction Hypothesis 19.1. The vector space of fields L which are tangent to order not exceeding n — 1 is of dimension n — 1 + 1 parametrized by Aiti, ..., An_ 1 ,1; Bn_ with the other constants defined by the formulae
2
¿,,2 = 0, A, = = 0, Ai,j = —g+2) Q ¿W+M, ~ 1) 1)
i-j+1 \j)
Bx
J + 1
(34j4n-2)
Aj+i,i,
or equivalently, the matrix of the coefficients of L is
An 0 0 0 •• • 0 -1 ¿2,1
¿2,1 0 0 0 •• • 0 -1 ¿3,1
, 0 -¿1,1 0 •• • 0 - 4 ¿4,1
¿4,1 0 -2^2,1 -¿1,1 •• • 0 - ! ¿4,1
An-2,1 0 An-i,i 0 00
ra-4 ( 3 )An-4,1 n-5 ( 4 M«-5,1
¿3 ) An-3,1 ) An-4,1
0
0
0
-(n - 3)^1,1 0
2 A Bn-1
2 ¿1,1
The goal is to show that similar expressions hold in dimension n. Thus, in coordinates (x1,. . . , xn—1,xn, «), we consider a hvpersurface, the equation of wh ich up to order n +1 reads as
2 2 n—2
ry*-' ry*-' ry /ry'lt'ry I ry ry \ ry °
__\ ^ / xm m— 1 \ ^ ^ \
u + — + + ^ 2(i — 1)!(j — 1)17+11 — 1)!
m=3 i,j>2 ^ ) \J ) \ )
i+j = m +1
rn~1T
xn— 1
2! 2!
1 ry . ry . ryft ry
I „n-2 V^ 1 __I x1Xn I 1
+ X1 Z.^ o ^ HI/„• IM + „I + X1
i,3> 2 i+j'=n
2 (i - 1)!(j - 1)! n!
£
i,3>2 i+j=n+l
1
2 (i - 1)!(j - 1)!
+ Oa
l(2) + Oa
and we consider a linear vector field
(n + 2),
with:
d d d d
L = X1 —--+ • • • + Xn_ 1 —--+ Xn—--+ U —,
ox1 oxn-1 oxn ou
X1 = ¿1,1 X1 +-----+ Ahn_ 1 xn_ 1 + B1 u,
Xn-1 = An-1,1 X1 + • • • + An- 1,n xn + Bn-1 u,
Xn An,1 x'1 + • • • + An,n xn + Bn u,l
U = C1 X1 +----+ Cn xn + Du.
Similarly, we consider tangenev of L to such hypersurface u = F(x1,
n, namely, the condition
which is equivalent to
0 = L (-« + F) I U=F) :
0 = <d(L - + F)| U=F).
xn_ 1,xn) up to order
(19.1)
XiXj
Letting Xn ■ 0 1(^)' applying the Induction Hypothesis 19,1, we see t
e matrix of coefficients of L involves known elements in dimension n - 1:
¿1 ,1 0 0 0 ••• 0 ¿1,n -1 ¿2,1
¿2,1 0 0 0 ••• 0 ¿2,n -1 ^3,1
¿3,1 0 — ¿1,1 0 ••• 0 ¿3,n - 4 ¿4,1
¿4,1 0 -2^2,1 -2^1,1 ••• 0 ¿4,n -1 ¿4,1
An-2,1 0 n14 (n~32)An~4,1 -2 (n-2\ a n-5 V 4 5,1 ' ' ' 0 An-2,n - An-1,1
An-1,1 0 n1 (n3 ) An-3,1 -2 (n- 1\ * n-4I 4 4,1 -(n - 3)^1,1 An— 1,n Bn-1
¿n,1 An,2 ¿n,3 ¿n,4 ' ' ' An— 1,n A -rin,n Bn
0 0 0 0 0 Cn 2 ¿1,1
Without letting xn := 0, by examining one can see that
A
1,n
An- i,n — 0, Cn — 0,
and the detailed verification of these vanishings is implicitly contained in the proof of the next Lemma 19,1, Hence, the matrix is
¿1,1 0 0 0 0 0 -1 ¿2,1
¿2,1 0 0 0 0 0 -,1
3, 0 -¿1,1 0 0 0 -1¿4,1
¿4,1 0 -2^2,1 -2^1,1 •• 0 0 -1 ¿4,1
An—2,1 An-1,1
0 0
— 1 (n—2\ A (n—2\ A
n-4 I 3 )An—4,1 n-5 I 4 )An—5,1
n—1 (V) An— 3,1 (V) An—4,1
-2 (n— 1
¿n,1 ¿n,2 00
¿n,3 0
¿n,4 0
0
-(n - 3)^1,1
An— 1,n
00
2 A - ^1 An-1,1
Bn- 1
Bn 2 ¿1,1
A R
and in order to complete the induction, we need to determine the values of ¿n,1; An,2, ¿n,3, A,nA, ..., An- i,n, An,n-, Bn-1, as follows.
Lemma 19.1. The tangeney condition (19,1) at order n + 1 forces the announced values
Bn-1 —--An,l, A.
n
n,2
A
k- 2
n,k
n - k + 1 \k
(3<k<n)
(k)
An—
n—k+1,1.
Proof. We abbreviate the hvpersurface equation (19,1) including independent and border-dependent monomials up to order n + 1 as follows:
0
u — F (x) + Ox/ (2) + Ox(n + 2),
and start to examine the tangency equation: 0 = L (-u +
'ind (L (-« + F)| u=F)
— ^ra+1 I
ind
= _ra+1
ra
(- U(x,u) + V X,fou) ■ (x) )
\ —' u=F (x)/
i=1
+ (2) + Oa(n + 2)) (19 2)
ra \
+ ^ Xt(x,Fn+\x) + Ox,(2) + Ox(n + 2)) ■ (Fnx+ + Ox,(1) + Ox(n + 1)) 1 i=i '
( - U [x,Fn+2(x)) + £ X, (x,Fra+2(x)) ■ Fnx^1(x)) , ^ i=i '
— ra+1
where we use the identity
0 = 1 (O * (1)) = (Ox(n + 2))
as well as the fact that Xi (^x,~Fn+1(x)^ are all Ox(1), hence multiplied by the last remainder Ox(n + 1) they become Ox(n + 2),
—n+2
In two steps, we want to apply Observation 18,2 to the products Xi • Fx. above for i = 1,..., n. First, each Xi is an Ox(1) and hence,
• Fnx+1) = (cjw • <d (K+1)).
Second, the vanishing orders at 0 of increase, as one can confirm by dilferentiating 19,1:
rn-U IP ' ind x\
„ra l p" '' ind I " X2
„ra l p" '' ind y " X3
■K
n I F
ind \ r xn—3
'K
n I F
ind ^ r xn-2
^ind ( ^
Xn — 1
„ra I p" '' ind \ " xn
X1 + x1x2 + ■ ■ + x 1 Xra
1! 2! (n - 1)!'
2 1 + 2 + ■ ■ + Tn—1T Xra—1
2! 1! 1! 1!(n - 2)!
ry3 1 + 3 x\x2 + ■ ■ + %n—2
3! 2! 1! 2! (n - 3)!
X
,n—3
(n - 3)!
„ra-2 1
ra 3 ra 2 ra 1
•Äy 1 X2 1^1 x 1 >Xj 4
(n - 4)! 1! (n - 4)! 2! (n - 4)!3!
(n - 2)!
rrra~ 1 1
^2 ^3
+
(n - 3)! 1! (n - 3)! 2! xra~1x2
(n - 2)! 1!
(n - 1)!
ra
1
n!'
here for later use, the vertical bar separates the pure monomials x\ from the monomials Therefore, applying again Observation 18,2, it suffice to know the independent parts of X1; X2, X3, ,,,, Xra—3, Xra—2l Xra-1, Xn up to decreasing orders:
ra+1 I V J?" ind lA1 ' t xi
ra+1 ra
= ^
ind
ra+1 1X2 ■ F,
X2
_ ra+1 / ra—1
= K-,
ind
ind (^hid(^1) ■ ^hid (f) ,
(<—^ ■ <d (rra +1)),
1
hid1 (X3 ■ Fl +) — ^d^S) ■ ) :
n1 ind
I Xn-3 ■ F
xn — 3
n+1 ind
( Xn-2 ■ F
Xn — 2
ind
n+1 1 X ! • F
1 r xn-i
n.
n+1
ind (Xn ■ Fx+ ^
n+1 1 — ^^ K:
(Vnd(^n- 3) ■ ^hid 1
(*fnd(^n-2) ■ ^d 1
^¡2nd(X^ 1) ■ ffnnd 1
(Vnd(^n) ■ ^ (Fn+1))
We recall that in all the X^ before tak ing ^n,d №) as above, we have to replace «by F (x) and for 3 ^ i ^ n — 2 this gives
Xi — Xi
(x,Fn +1(x))
j - 2
x) ) — ¿¿,1^1 - E . ^ . + i Q) A-j+1 Xj - ^.+1,1 ^n+2(x),
2
ï +1
"Rn+2,
with similar formulae for i — 1, 2,n - 1, Fortunately, for all 3 ^ j ^ i, the values of the coefficients of the Xj will be unimportant, hence we shall abbreviate:
n—i+1
ind
2
(Xi) — Ai,1 X1 - * x3-----* Xi - —— ¿¿+1,1
i + 1
X1
"2T
+
x\x2
HT
++
Xn—i
(n - z)l
In Xn, there is also no contribution of Bn u = Bn ~Fn sinee ~Fn — OX(2):
^ind (Xn) — ^nd An,1 X1 + ■ ■ ■ + An,n Xn + Bn Ox (2)
¿n,1 X'1 + ■ ■ ■ + An,n Xn.
Finally, we can write in length all the terms of the tangenev equation (19,2), as follows without mentionining ^"d1^):
0 =
(22 n ry2 ry ry ry' ° ry
X1 + X1X2 + ■ ■ ■ + X1 xn
(
2T 2T
nl
)
+ ( ¿1,1^1 - ^¿2,1
2 1
L2T
2
+ X^X2 + + Xn Xn— 1
2T
(n - 1)l -j
2 n — 2 n — 1
ry2 ry ry 0 ry ry 0 ry
+ .Ay 1^2 + + 1 n— 1 + n
11
+ ( ¿2,1X1 - 3¿3,1
X1
L2Î
(n - 2)1 (n - 1)1
2 n—2
+ X1X2 + + X1 Xn-2
21
(n - 2)1 J
2
n_2 n_1
+ ^1^2 + + X1 Xn— 2 + X1 Xn— 1
1111 11 (n - 3)1 11 (n - 2)1
+ ( ¿3,1^1 - *^3 - 2¿4,1
X1
L2"
2 n_3
+ X1X2 + + X1 Xn— 3
21
(n - 3)1 J
>
3
n_2 n_1
+ ^1^2 + + X1 Xn— 3 + X1 Xn— 2
2111
21 (n - 4)1 21 (n - 3)1
+ ( ¿¿,1^1 - *X3-----*Xi - t-^T¿¿+1,1
1 + 1
X1
L2T
ryn 1 ry
X 1 Xn — 1.
X1X2 ^n—i
+ ^T + ■ ■ (n - ¿)1 j
)
1
1
2
1
3
1
kL> 1
+
Of ^ Of -
X1X2
(i - 1)111
++
X
,n—2
Xn—%
+
X
,n— 1
Xn—i+1
(i - 1)1(ra - i - 1)1 (i - 1)1(ra -
¿+1_
- 01/
+ An-3,1 X1 - *X3-----*Xn-3 -
n- 2
An—
n—2,1
2 1
21
23
rf2 Of n rf rf ^
(n - 3)1 + (n - 4)111 + (n - 4)121 + (n - 4)131)
^^ 3 1
^n 2 X3
x 1 x4
+ An-2,1 X1 -*X3-----*Xn-2 -
n— 1
■ An—
n—1,1
X1
L2Î
+
21
2
31
)
)
_n-2
_n-2,
„n— 1,
_ + Xj '12 + Xj 'X3 \
(n - 2)1 + (n - 3)111 + (n - 3)121y
+ An-1,1 X1 -*X3-----*Xn-1 + Bn-1
2 n
X
L21J
)t
n— 1
+
X'
,n— 1
^2
(ra - 1)1 (n - 2)111
)
+ (¿n,1x1 + ■ ■ ■ + An,kXk + ■ ■ ■ + An,nXn) ( 1
X1
m
We know by the induction assumption that ^.dM applied to this gives zero. Thus, when applying it suffices to collect all independent m0nomials of order n + 1, namelv x'-x1,
and to determine the coefficients of these monomials, which should all vanish. Let us explain how to determine the coefficient of a general 'intermediate' monomial x\Xk with 2 ^ k ^ n — 1:
• the line i = n — k contributes two terms;
• the line i = n — k + 1 contributes one term;
• the line i = k contributes one term; and that is all. Therefore, we obtain:
k
XxXk
(
2
n — k + 1
An—
n-k+1,1
1
+
2 (n - k - 1)1(k - 1)1 k1(n - k)1
+ An~
n-k+1,1
+ ^ A.
(n - k)1(k - 1)1 n1
n,k
The coefficients of x-x1 and 0f x-x2 can be determined directly from what is written as well
as the coefficients of xr^xn—1 and of xnxn-
0 = x
n+1 l 1 1 y 21
1 1 1
7-777 Bn-1 +--T An,1
(n - 1)1 n\
)
+ X^X2
21
1
11
+
n - 1 21 (n - 3)111 n - 1 21(n - 2)1 (n - 2)111 J
An-1,1 +--T An,2
n\
)
+
k
+ X1 Xk
+
n - k + 1 21 (n - k + 1)1(k - 1)1 n - k + 1 k\(n - k)1
+
(n - k)1(k - 1)1
An-k+1,1 + T An,k n\
+ Xn Xn— 1
21
1
1
+
1
2 21 01(n - 2)1 2 (n - 1)111 11(n - 2)1 J
¿2,1 +--T An,n-1
n!
)
1
1
2
1
1
1
1
1
2
1
+
(
1
- 2— +
1
n! (n — 1)!-
+ ^ a
n!
)
So, we obtain the announced values for Bn-1, ¿n,2, ..., An,k, . In conclusion, the induction on the dimension n is complete.
A -, A
-rin,n— 1) -rin,n-
□
20. Infinitesimal action at order n + 2
Beyond dimension n ^ 6 (Section 15), we have no explicit formula for the subgroup of GL(n + 1, R), which stabilizes the normalization (19,1) up to order n +1 and which would be valid for each n ^ 2, We will therefore proceed in an infinitesimal manner. This will be less expensive, computationally speaking.
Thanks to Section 19, we can take a vector field tangent to (19,1) up to order ^ n +1:
1+1
L = ( ¿1,1 Xi — ¿2,1 u^ dXl + ^¿2,1 Xi — —¿3,1 u ) dX2
2+1
+ ¿3,1 X1 + 0 + ¿3,1 X3 —
+
2
3 + 1
¿4,1 « d.
JX3
+ ( ¿n-1,1 X1 + 0 + ¿n-1,1 X3 +-----+ ¿n-1,n-1 Xn-1 — - An,1 u ) dxn_i
+ (¿n,1^1 + 0 + ¿n,3 ^3 +-----+ ¿n,n-1 Xn-1 + An,nXn + Bn u) 3,
+ (2 ¿1,1 u) du,
where, for all 3 ^ j ^ i ^ n,
A
,
U — 2)
i — 3 + 1 \j
(! )
With small e « 0, this L has an approximate flow: y1 = X1 + £ ^¿1,1 X1 — Y+y ¿2,1 u^ + O(e2),
y2 = X2 + £ ^¿2,1 X1 — ¿3,1 u^ + O(e2),
y-3 = X3 + £ ^¿3,1 X1 + 0 + ¿3,1 X3 — 3+y ¿4,1 u) + O(^2)
yn-1 = Xn-1 + £ An-1,1X1 + 0 + ¿n-1,1 X3 +-----+ An-1,n-1Xn-1--¿n,1 u
n
)
(20.1)
(20.2)
(20.3)
+ O(e2),
yn = Xn + e (¿„,1X1 + 0 + An,3 X3 +-----+ An,n-1Xn-1 + An,nxn + Bn u) + O(e2),
v = u + £ (2¿M u) + O(e2).
For 1 ^ i ^ n — 1, we can write the intermediate lines as follows:
yi = Xi + e ^Ai,i X1 + 0 + ^ Aij Xj + 0 + • • • + 0 — ¿¿+M u^ + O(e2). (20.4)
With its independent monomials of order n + 2, the hypersurface equation on the left is
2 2 n m 1 = -H + X1X2 + ^^ /X1 xm + _m-1 1 XiX3_ \
U =2+2 + m\ +X1 2(i — 1)!(j — 1)17
m=3 ¿,j>2 ^ ' ^ '
i+j = m +i
+ F+2n n x"+1x1 + F+n n X2 + ••• + F +1n 1 Xra (20.5)
+ 1n+2,n,...,n, 0s, + 1 n+1.1.....0 / \i + + 1 n+1,0,...,1 / i 1 M v 1
(n + 2)! (n + 1)! (n + 1)!
n \ ^ 1 XiXj
+ E 2«- 1W -1)! + °v(3) + O'(" + 3)-
i,3>2 i+j=n+2
Similarly, the hypersurface equation on the right is
2 2 n m 1
^ = ft + № + V^ fy^m , „.m-1 V- 1
E (tt+11 E
2 2 ^ V m! y1 ^ 2 (i - 1)!(j - 1)!7
m=3 2 ^ ' '
i+j = m +1
, rr ^V , ^ , , ^ ^ra+1 /90 6)
+ ^ra+2,0,...,0 7-—7TT + ^ra+1,1,...,0 7-—¡TT + + ^ra+1,0,...,1 7-—¡TT ^U.kjj
(n + 2)! (n + 1)! (n + 1)!
1 ViVj
i+3=n+2
+ si E 2(irrÜTI)! + °V(3) + O(» + 3).
i,j>2
Now we assume that (y1,..., yn, v) are replaced from (20.3) in 0 = — v + G(y), that u is replaced by F(x1,... ,xn) from (20.5):
0 = E (x1,...,xn, e) = E (x, e). We consider the terms of orders not exceeding n + 1:
n +( Eai.....an (e)x\1 • • • xnj = Eai.....an ^^ • • • Xn •
o"iH-----|-<Tn^ra+1
Lemma 20.1. The fact that L is tangent up to order n +1 implies
0 = nn-1 (—v + G(y)) = nn-1 (E(x, £)).
We look at order n + 2 terms:
0 = (E(x, e)) = nn+2 (E(x, e)) =: En+2fl,...,n(e) x'i-1X1 + Era+M.....n(£) XI^+1X2 + • • • + E^n.....^) xl^+1xn.
For e = 0, the map is the identity onr, hence without computation we know that
0 = E ( — ^ + ^ + * T,.....+ 2)) + .....„ (» + 3),
vi+--------- =n+2 N 7
and our key goal is to determine the terms TVl...,Vn that are of order 1 in e, more precisely, to compute the independent ones:
Tn+2.n.....n, Tn+1.1.....n, • • • , Tn+1.n....,1. (20.7)
Equivalently, we can write the fundamental equation by specifying the dependent monomials
as the remainder O
X2,...,X,
(2)
0 -E (x, e)
--I
Fn+2,0,...,0 Gn+2,0,...,0 ^
+ ——T^T-i" + £Tn+2,0,..,0 + O(e )
(n + 2)! (n + 2)!
)
+ Xni+1X2[ — Fn+1:1,::,0 + Gn+1,1,...,°
(n +1)! (n +1)!
+ £Tn+1,l,...,0 + O(e2)
(20.8)
+
+ xn1+1xj — + + e Tn+1,c,...,1 + O(e2)
(n +1)! (n +1)!
+ OX2,...,Xn (2) + OX1,X2,...,Xn (n + 3).
We introduce an operator
nn+2\de
s=0
Xn+1X
+ xn p
+ ~ TTT r n
which selects what we want to compute:
i-ind / 7~r/ \\ Xn ^^ nn+2 (Ee)) = 7-—rn+2,0,...,0 + 7-—¡1 rn+1,1,...,0 + ' / .
+ (n + 2)! (n + 1)! (n + 1)!
Also, for studying the remainders, we shall need to consider all independent monomials of order not exceeding n + 2:
n+1,0,...,1.
nn+2(.) := —
Uind W := de
£=0
n+2
ind
(•)
n+2
ind
/ d
=0
(•) .
Thus, we should apply nn+2(^) to all (numerous) terms of (20,6), We start by the remainders. Lemma 20.2. The identity
£=0
(Oy2...„yn (3)) = Ox2,...,x„ (2)
holds.
Proof. Take any monomial y^2 ■ ■ ■ y-—n with u2 + ■ ■ ■ + un ^ 3, and abbreviate (20,3) as:
Vi = Xi + e Ri(x) + O(e2), 2 ^ i ^ n,
whence,
[X2 +£R2 + O(£2)Y2 ■■■(xn + eRn + O(£2))
= fa2 + //2 x?'1 £ R2 + O(e2)) ■ ■ ■ {x- + Un x-"-1 £ Rn + O(e2))
— ^"2 . . . + c
>Xj 2 n ^
U2XV22-1R2XI3
. + . . . +
•A'— I I ^2
■ TVn-1u TVn-1 R xn— 1 ^n^n n
+ O(e2),
□
and here u2 — 1 + u3 + ■ ■ ■ + un ^ 2, ..., v2 + ■ ■ ■ + un-1 + un — 1 ^ 2 as well. Thus, the remainder Oy2,...,y„(3) in (20,6) has contribution equal to 0 in n—+2(E),
n
Next, still in (20,6), we consider border-dependent monomials from the sum > but only
m=3
for m ^ n — 1 at first.
Lemma 20.3. For every 3 ^ m ^n — 1 and for a 11 2 with i + j = m +1, we have
0 = n—n+2 (yr1 ViV3) .
Proof. To simplify (20,4) after replacement of u by F, we abbreviate
A.
ai := -
i+1
z + 1 (n +1 — m)!
and use • • • to denote monomials of order not exceeding n — m +1:
Hi Xi + s
+ e i Ai,i + ^ A
^ 3<Ki
2
i,3 Xj
Ai
i+1,1
+
X
n+1-m 1
Xn+1-m
(n +1 — m)!
2 + 1 + O(e2)
Xi + £ { • • • + aiXn+1-mXn+1-m + Ox(n — m + 3)} + O(e2).
+ Ox(n — m + 3)
}
We write a product
ym-1 ViVj = (x1 + { • • • + a1 Xn+1-mXn+1-m + Ox(n — m + 3)} + O(e2))m"
• (^Xi + £ { • • • + aiXn+1-mXn+1-m + Ox(n — m + 3)} + O(e2)^j
• (x¿ + £ { • • • + aj xn+1-m Xn+1-m + Ox(n — m + 3) J + O(e2)^j
and we expand:
^n+2 {y? 1 ViVj) = ^n+2 (xm 1XiXj \ + e Ixm 2 (m:1) a1 xn+1 mXn+1-mXiXj
,m-1 ra+1-m
++ x m ai x 1
Xn+1-m Xj
+ Xm Xi aj Xn+ Xn+1-m| + O(£ ).
To complete the proof, we observe that XiXj are dependent monomials, and since n — m +1 ^ 2, observe that the two monomials xn+1-mXj and xn+1-mXi are also dependent, □
It therefore remains to compute
n
ind
n+2
(
— v + ^1 +
2 2 3
y2! , y 1iJ 2 , ytya
2! 2!
+
3!
+ • • • +
n- 1 n- 1 n n 1
(n — 1)! + n!
+!>r1 E
yiyj
i+j = n + l
2 (i — 1)!(j — 1)!
+ r yni+lyi + r yn+^2 + + r yn+1 yn
+ ^n+2,0,...,0 7-, P.S, + ^n+1,1,...,0 7-TTVT + • • • + ^n+1,0,...,1
+^n E
(n + 2)!
1 yiyj
i+j=n+2
2 (i — 1)!(j — 1)!
)
(n + 1)!
(n + 1)!
Since n™+2(^) is linear, we can proceed termwise. We compute the first term
nn+2(— V) =
"n+2 I
(-
\ de e=0\
— u — e 2 A1A u + O(e2)
))
= ^n+2( — 2 ¿1,1
ÍY>2 /y>2 ry> ~ rpn rp rpn+1 rp
2Í + 2! + ''' + + Fn+2,0,...,0
+ Fr
n+1,1,...,0
^n+1^2 (n + 1)!
+ • • • + Fn
Xn+1x,
n+1,0,...,1
1 n
(n + 1)!
+O
xi,...,x,
Xn + 3)
2
1
1
and we obtain:
nind (
n n+2 (
—v) — -
22
-,-—U Fn+2,0,...,0 ¿1,1 x'l - 7-—¡T7 F,
(n + 2)1 (n + 1)1
n+1,1,...,0 ¿1,1 ^n+1
X2
2
(n + 1):
¿1,1 xn+1
(20.9)
Xn
Next, for k G N, abbreviating
^ : OX2,...,X„ (2) + OX1,X2,...,X„ ( k),
we compute the second term:
nn+2 I ~
(f) — ^f 2 fc + ^1,1*1 - 1^2,1
lT
+ ■ ■ ■ +
>^n—1
(ra - 1)1
f- + + Ox (2) + Ox(n + 2)]} + O(e2))2)
n
ind
n+2 I
2 2
/V»n /V»
Xn
-^¿2,1^ + nn+^ + O(e2))'
>n+2\
m
1
we obtain:
nind I
nn+2 1 -
d) — 221 (-1-21)
n
1n
n1
1
A
--T ^2,1 X1
n1
n+1
Xi
(20.10)
We treat the third term:
2 2 2
nind ' -- i
2 )
X2 + £
- 2 ¿2 1 ^ 1Xn—,1 + 'Rn+1
2 2,1 (ra - 1)1
2 Tn~1T
- 3 ¿3,1 T^)1 + ^n+1
3 (n - 1)1
+ O( 2)
+ O( 2)
>0
that is,
n
ind
n+2
(V2iV2\ 12 I 2 A \ 1Xn-1
{-2T) — 2*2 l-a^J "(17-1)7
2
1
A3Axn+1Xn+1. (20.11)
3 21 (n - 1)1
Similarly,
nind f yhjA 1 n+2 ^ 31 )
n-d^ + £
a (
(x3 + £
2 Tn~2T - 2 117-2)7 + n
2
----- ¿4,1
xn—2
4 4,1 (n - 2)1
+ K1
+ O(£ 2))
X 1
O( 2) 1
that is,
nind
nn+2 \ 31
(V3lV3\ — 1 T3( 2 A \ ^2^n-2
V"3rJ — r4^7 117-2)7
4 31 (n - 2)1
+
¿4 ,1X1 X-n-2.
Now we consider general m wit h 3 ^ m ^ n - 1
n
ind ( VTVm A
n+2V m\ )
(1-,
Û(Xl + £
2
- 2 1 ^_xn-rn+1 + ^n-m+3
2 2,1 (ra -m +1)1
+ £
2
m + 1
A
x
m+1,1
,n—m+1 1
(n - m +1)1
^n-rn+1 + j^n-m+3
)m
+ O(£ 2))1):
2
3
2
1
that is,
nind
n ra+2
m! m! 1 -
m
A \ X1
(
■n—m+1^.
■¿ra—m+1
(n - m +1)!
¿m+1,1 X^ Xra—m+1.
m +1 m!(n - m + 1)!
For m = n, the result is different, two monomials are obtained:
nran+2 (i vravn)
nran+^(x1+e
■■■-2a2,1 |+n3
+ O(£ 2))
+ Bra -1 + n3
+ O(e2)
■ (xra + I
S | + a (ra) <—1 (-
(20.12)
that is,
nind ( —
1
Bra X^
1
ra!2!"ra~1 (n - 1)! 2! Next, for all i, j ^ 2 satisfying i + j = n + 1, whence i,j ^ n - 1:
¿2 ,1 %ra.
n
ind
ra+2
(yra—]1
1
ViVj
)!
1
1
2 (2 - 1)!(j - 1)!7 2 (2 - 1)!(j - 1)!
nind
nra+2
(20.13)
Xi
■■■-^2,1^ +113
- i2 1 2 2,1 2
¿i+1,1 ^ + n
+ O(e2))
ra1
ij+1,1
+ O(e2
X1
¿,+1,1 y + n
+ O(e2)
1
1
2 (i - 1)!(j - 1)! 1
X1
1 (-ih*»")
1
+ t
2 (i - 1)!(j - 1)!
™ra— 1 1
2
X1
1
¿7+1,1 I — x
that is,
nind
nra+2
(V1
2 (i - 1)!(j - 1)!
!
1
1
2(2 + 1) (2 - 1)!(j - 1)! 11
*i+1,1 x'ra+1X
2(j + 1) (i - 1)!(j - 1)!
Aj+1,1 X™+1Xi.
In fact, we need to find a sum:
nran+2 yra
ind (V;1 E
ViVj
j>2 i+j=n+1
2 (i - 1)!(j - 1)!
\ 1 ra—1
= - 2 £
/ 1=2
2 n - i + 2 (n - j)!(j - 1)!
¿ra—j+2,1 x1+lX
ra— 1 £
n -i + 2(i - 1)!(^ - «)!
TT *ra—i+2,1 X™+1X.
2
2
1
2
2
1
1
1
1
1
1
and by exchanging j <—y i in the first sum, the two sums are equal:
n 1
n
(î/î—1 L
ind /„ ,n—1
n+2
1
ViVj
i,3> 2 i+j=n+1
2 (i - 1)1(J - 1)1
o — - E
¿=2
1__1_ A
n -i + 2(i - 1)1(ra - l)1An—+2,1X1
(20.14)
The next term is
y1+2
G.
nind I G y1 \ — Gn+2,0,...,0 nind . . + _
nn+2\ Gn+2,0,...,0 ^ , — ^ , n-+2 I 1^1 + £
has value
(n + 2)7 (n + 2)1
Gn+2,0,...,0 ^n+2
(n + 2)1
yn+2
¿1,1X1 + ft2 \ 1 ) X1 ^1,1 X1
+ O(e2
n+2
nn+2(Gn+2,0,...,0 (^+2)1
Gn
(n +1)1
¿1,1^
(20.15)
Then
n„+2 G
,,n+1_
yr ^2 \ — Gn+1,1,...,0 nind
n+^Gn+1,1,...,0 (n +1)^ — (n +1)1 nn+2
X1 + £ X2 + £
¿1,1X1 + ft2
¿2,1 X1 + ft2
+ O(£ 2))n+1)
+ O( 2)
Gn+1,1,...,0 n+1 ^ 1 Gn+1,1,...,0 n m+1\ A —- s. X1 ^2,1^1 + -,-. -, M ( 1
(n + 1)1 (n +1)1 1
that is,
ind
n n+2 I
,,n+1.
(Gn+1,1,...,0 f^J^) — ¿2,1 x-+2 + ¿1,1 x^1 ^ (20.16)
V (n +1)1/ (n +1)1 n\
G
(n +1)1/ (n +1)1
Next term is
G
,,n+1
n+HGn+1,0,1,...,0 (n +1)7— (n +1)1 n-+2
(
¿1,1X1 + ft2
¿3,1^1 - ¿1,1^3 + ft2
+ O(£ 2))n+1)
+ O( 2)
G
that is,
nl+2 ( Gn+1,0,1,...,0
y1+1y3
(n +1)1
Gn+1,0,1,...,0
n+1,0y{^n+1 ¿3,1 X1 - X-+1 ¿1,1 X3 + x- (n+1) ¿1,1 X1X3}
f {¿3,1 ^n+2 + T^) ¿1,1 ^n+1 X3}. (20.17)
(n +1)7 (n + 1)! I 3,1 1 (n +1)! Then for general k with 3 ^ k ^ n, we compute in 0 ■ ■ ■ 1 ■ ■ ■ 0, the 1 is at position k — 1:
n
ind
n+2 !
(G y'1+lyk \ — Gn+1,0-1-0 nind /V +p
lGn+1,0-1-1 (7TÏT ) — (n +1)1 "n+H^1 +£
(n + 1)1/ (n +1)1 ■ (^k + £ Gn+1,0-1-0
(n +1)1
Gn+1,0-1-0
(n +1)1
¿1,1X1 + ft2
+ O( 2)
\ n+1
Ak,1x1 + E Akjxj + ft2 +O(e2)
3^Kk
X-+1 (¿k,1 X1 + E Aktj x3) + xn (n+1) ¿1,1 X1Xk 1 V 3<Kk / J
¿k,1 ^i"+2 + E ¿kj
1+1
X f'
3<Kk-1
+ (Aktk + (n + 1) AM) xnl+1xk
.
Now we replace the Ak.j by their values from (20,2),
Ak,k = — (k — 2) ¿1.1
and we obtain:
\ 1
(n + 1)U (n +1)!
nind (G V1 Vk \ = 1 G A ~ra+2
nra+2 ^Gra+1,0-1-0 (n + 1)^ = , + 1)! Gra+1,0-1-0 Ak,1
E1 J 2 fk\^ A ra+1
7-TTTT 1-• , -t (J Gra+1,0-1-0 Ak—j+1,1 Xj
3<,-<k—1 (n + 1)! k - j + 1 ^ , J , 1
+ n -k + 3 G ^ra+1
(n + 1)!
(20.18)
Finally, for all i, j ^ 2 with i + j = n + 2 we have
^fa - $$ - 1),) = 2« - 1)!b- - 1)! ^(fc + * [Au*1 + n2] + O(a)"
■ (xt + e\Az,1X1 + E AhkXk + n2] + O(e2))1 V L -1 /
■ + e \Aj,1 X1 + E A^ Xi + n2! + O(£2)) )' v L -1 / )
that is,
nind A ra 1 \ = 1 _1_ A -.ra+1^ + 1 _1_ A . Tra+1T.
iira+2^ 2(i - 1)!(j - 1)!7 = 2(2 - 1)!(j - 1)! A%,1 1 x3 + 2 (i - 1)!(j - 1)! ^^ ^ We find a sum:
nind (7 ra V^ 1 ViVj \ = - 1 ^ _1_ A xra+1
JL 2(2 - 1)!(j - 1)^ = 2 ^ (n - j + 1)!(j - 1)! Ara-.?+2,1 xi XJ
i+j=n+2
1 ra 1
1 y 7-777^-77 Ara_,+2 1 Xra+1X.
2 ^ (2 - 1)!(n -2 + 1)! +2, 1
i)
and by exchanging j <—> i in the first sum, the two sums are equal: ( \ „
nind
nra+2
1
Z 1)! = (,-_1)!(.1_,- + 1)! A^^+2,1Xra+1x^. (20-19)
v ^ - W - 1)!,
\ i+j=n+2 J
1
=2 (2 - 1)!(n -2 + 1)!
j=2
We sum all terms (20.9), (20.10), (20.11), (20.12), (20.13), (20.14), (20.15), (20.16), (20.17), (20.18), (20.19):
22
nra+2 (-V + G(y)) = - --—— Fra+2,0,...,0 A1,1 X^2 -7-—77 Fra+1,1,...,0 AM <+^2
(n + 2)! (n +1)!
21
Fra+1,0,..,1 A1,1Xrl Xra - —~.A2,1Xrl X„_
/ . 1 \ i " ra+1,0,...,1 " *1,1 I ^ra 1 A2
(n + 1)! n
1 1 A ~ra+1_ _ o / i\i A3,1 ^ra— 1
3 (n - 1)!
j ^ ¡7 1 7VT Am+1,1 X1 Xra-m+1 -
m +1 m!(n - m +1)!
2 1 1 1
¿n,1 %2 + TT Bn X- - . ¿2,1
n (n — 1)!2! 1,11 21 n!2 -1 1 (n — 1)!2 n-1 1 1
\ " 1__1 A „ ,lTn+1T.
/, •lOi'' i\l/ -M ^n+2-t,1x1
2-' 11. — 1+7 I 7 — Mil 11. — 7)1
n — i + 2 (i — 1)!(n — ¿)!
+ 7-. -i \. Gn+2,0,...,0 ¿1,1 ^n+2 + 7-, 1 s . Gn+1,1,...,0 ¿2,1 ^n+2
(n +1)! (n +1)!
+ T Gn+1,1,0,...,0 ¿1,1 Xr—X2
n!
+ Z iTIT-r-
777 G—+1,0-1-0 ¿k,1
1
( ra + 1)!
3<k<n
1 J — 2
^ (n + 1)! k — — + 1 ® G—+1,0-1-0 ¿k-^+1,1
3<Kk-1
+ ^-7T7" Gn+1,0.
(n + 1)! J
n 1
+ / y ~t- im/ : i 7VT ¿n-i+2,1 ( z — 1)!(n — % + 1)!
j=2
and collect the coefficients of the independent monomials: 2 i 2 1
0 — X1 1 — 7-. 0m ^n+2,0,...,0 ¿1,1 + 7-. m Gn+2,0,...,0 ¿1,1
[ (n + 2)! (n +1)!
1
(n + 1)!
+ ^ (7TT)i G—+1,0-1- ¿k,1 + ¿^j
21
+ £2^ — 7-TTTT ^n+1,1,...,0 ¿1,1 +--r Gn+1,1,...,0 ¿1,1
1 (n + 1)! n!
2 1 . 1 1 . 1 1 . 'n (n — 1)!2! 1 n 1!(n — 2)! 1 + 1! (n — 1)! ^
}
_ [-2-(n-1)2+n-2] = An,i ^ [=2+2^] = '
1
+ —7—^7 ^n+1,01-0 ¿1,1 + ^—3 + ,3) Gn+1,01-0 ¿1,1 + * ¿2,1 +-----+ * ¿n-2,1
1 (n +1)! (n +1)!
. r__^ 1___^ 1 1 n 1
+ n-1,1^ n — 1 (n — 2)!3! n — 12!(n — 3)! + 2!(n — 2)!
= ^"-1,1 (n—1)!3! [-2-(n-2)3+(n-1)^ = A„_1,1 (n_1i),3, [-2+3]
+ x—+lxA — 7-Fn+1,001-0 ¿1,1 + -r+T Gn+1,001-0 ¿1,1 + * ¿2,1 +-----+ * ¿n-
1 x4 S — 7-TTTT rn+1,001-0 ^1,1 +—}-TTTT G—+1,001-0 ^1,1 + * ^2,1 + ■ ■ ■ + * ^n-3,1
(n + 1)! (n + 1)!
. r__^ ___^ 1 1 n ]
+ n-2,^ n — 2 (n — 3)!4! n — 2 3!(n — 4)! + 3!(n — 3)!
= ^«—2,1 ^^ [-2-(n-3)4+(n-2)4] = An—2,1 ^^ [-2+4] + — 7-Fn+1,0001-0 ¿1,1 + -5+T Gn+1,0001-0 ¿1,1 + * ¿2,1 +-----+ * ¿n-
1 — 7-777 rn+1,0001-0 ^1,1 +--}-TTTT Gn+1,0001-0 ^1,1 + * ^2,1 + ■ ■ ■ + * ^n-4,1
(n + 1)! (n + 1)!
A r__^ 1___^ 1 1 n 1
+ ra—n - 3 (n - 4)!5! n - 3 4!(n - 5)! + 4!(ra - 4)!
= A»-3.1 [—2—(ra—4)5+(ra—3)^ = An-3.1 [—2+5]
+
„ra+1, I 2 * . n-k + 3
+ xU - --—— Fra+1,0-1-0 A1,1 + —-—— Gra+1,0^1^0 A1,1 + * A2,1 +-----+ * Ara—k+1,1
1 (n + 1)! (n + 1)!
+ Ara— k+2,1 ^-u, ovu I [ - 2 + k] j
(n - k + 2)!k!
+
24
+ X\ Xra—1\ - 7- s. ^ra+1,0-10 A1,1 + 7-. M Gra+1,0-10 A1,1 + * A2,1
1 (n + 1)! (n +1)!
+ A3,1^\Ü~Y\\ [ - 2 + ^ - 1]}
3!(ra - 1)! 23
+ - ---^+1,0^01 A1,1 + 7-TTTT Gra+1,0—01 A1,1
(n + 1)! (n + 1)!
1 , , ,
+ A2,1 [ - d - (n - 1)!2! - 0 + (n - 1)!1!^ }.
= A2,i da [-2-™+2"] = A2,i M
With this expression, we have therefore computed the quantities Tra+2.n.....n, Tra+1.1.....n, Tn+1.n.1.....n^ ■, Tn+1.n.....1 introduced in (20.7).
21. Normalizations of order (n + 2) monomials
Coming back to (20.8), thanks to the above expressions of Tra+2.n.....n, Tn+1.1.....^ Tra+1.n.1.....n, ■ ■ ■, Tra+1.n.....^, the coefficients at x™+2, xrl+1x2, xrl+1x3l ..,, x™-1 xn should vanish and hence, for the first three of them we obtain:
0 = - (nT2)i Fra+2,0,...,0 + (^+2)! Gra+2,0,...,0
A1 ,1 + 7-, 1 M Gra+2,0,...,0 A1,1
(n + 2)! 2 " (n +1)!
+ Z (räTT)i Gra+1,0-1-0 Ak1 + ^HBra\
0 = - 7-^ra+1,1,...,0 + 7-! 7TT Gra+1,1,...,0'
(n + 1)! (n + 1)!
0 = - 7-^ra+1,0,1,...,0 + 7-! 7TT Gra+1,0,1,...,0
(n + 1)! (n + 1)!
2 n-3 + 3
^ra+1,0,1,...,0 A1,1 +---r™rvr Gra+1,0,1,...,0 A1,1 + * A2,1 +
(n + 1)! "+1,0,1,...,0"1,1 ' (n +1)!
-2 + 3
}
+ * ¿n-2,1 + (n _ 1)!3, ¿n-l,l + O(e2) and for general k with 3 ^ k ^ n _ 1 we get
0 = _ 7-Fn+1,0-1-0 + -—- Gn+1,0^1^0
(n + 1)! (n + 1)!
2 n _ k + 3
Fn+1,0-1-0 ¿1,1 +--/-TTTT Gn+1,0-1-0 ¿1,1
(n + 1)! 74+1,0 • • 1 • • " 1,1 ' (n +1)!
_2 +
+ * ¿2,1 + + * ¿n-fc+1,1 + (n _ k + 2)!k! ^™-fc+2,1
}
and for the last two we find
0 = _ 7—^TT ^n+1,0,...,1,0 + 7—Gn+1,0,...,1,0 ( n + 1)! ( n + 1)!
2 4 _2 + n _ 1 Fn+1,0,...,1,0 ¿1,1 + 7—TT77 Gn+1,0.....1,0 ¿1,1 + * ¿2,1 +
i I I n+1,0,...,1,0 - '1,1 I / ,11 "n+1,0,.,.,1,^'1,1 I ■ ' 2,1 I 0I / TM
( n + 1)! ( n + 1)! 3! ( n _ 1)!
+ O(£ 2),
0 = _ (nTT)i Fn+1,0,...,0,1 + (n+T^ Gn+1,0,...,0,1
1
2 , 3 _2 + n , ,
)
+ £ i — T-TTi En+1,0,...,0,1 ¿1,1 + 7-TTTT Gn+1,0,...,0,1 ¿1,1 +--77—r~ ¿2,w + O(£ ).
(n + 1)! (n + 1)! 2! n!
We are going to solve the G^ in terms of the F^. For some v = (u1,..., un) G N— we consider an equation with constant a, /3, A,:
0 — — Fv + G, + e{aFu + / G, + A, To determine the term Sv with
G, = Fv + eSv + O(e2),
we replace and identify
0 = _FV + Fv n + eSv + O(e2)+e{aFu + ß (Gv + e Su + O(e2)) + A,} + O(e2)
— e {S, + (a + /) G, + A,} + O(e2),
and hence,
S, := — (a + /)GV — A,.
Formulae of the same kind exist for a linear system involving several Fv, G, as above. We therefore obtain:
G—+2,0,...,0 = ^n+2,0,...,0 — epfn+2,0.....0 ^1,1 + (n + 2) Fn+1,0-1-0 ¿k,1 + {rt+2)Bn> +O(£2),
I 2<k<n
Gn+1,1,...,0 = ^n+1,1,...,0 — £ \(n — 1) Fn+1,1,...,0 ¿1,1
2+3
Gn+1,0,1,,...,0 = Fn+1,0,1,...,0 — e {(n — 2) Fn+1,0,1,...,0 ¿1,1 + * ¿2,1 +-----+ * ¿n-2,1 + 7-7777:7 (n +1)! ¿n-1,
(n — 1)!3!
}
2
for general fc wit h 3 ^ k ^ n — 1 :
Gn+i,o-i-o = Fra+i,o^i^o — £ { (n — k + 1) Fn+i,o-i-o Ai,i + * A2,i + ■ ■ ■ + * An-k+i,i
[
}
— 2 + k
+ (n — 2 + 2)!fc! +1)! +O(f2)-
and
—2 + n — 1
Gn+i,o,...,i,o = ^n+i,o,...,i,o — £ { 2 Fn+i,o,...,i,o ¿1,1 + * ¿2,1 + 3!(rc — 1)! (n + 1)! A3,i} + O(e ),
1
I
—2 + n
— £< 1 Fn+i ,o,...,o,i ¿1,1 + (n +1)! A2>i} + O(e2).
It is more natural to begin with the last equation.
Lemma 21.1. It is possible to on the right, and then on the left:
Gn+i,o,...,o,i := 0, F.n+i,o,...,o,i := 0,
Gn+i,o,...,i,o := 0, ^n+i,o,...,o,i := °
Gn+i,o,i,...,o := 0, Gn+2,o,o,...,o := 0,
^n+i,o,i,...,o := 0, ^n+2,o,o,...,o := 0,
while the coefficient of x^+lx2 is a relative invariant:
G.
n+1,io-o
oc F,
n+1,io-o.
Proof Indeed, the variable A2,1 is free to make the first normalization, then the variable ¿3,1 is free to make the second one, and so on until Bn is free to make the last normalization. We also observe that the system is triangular. Two other views of these normalizations will be given in the next two Section 23 and 22, □
22. Alternative more direct normalizations at order (n + 2)
As we know from Section 19, the matrix of coefficients of a vector field L tangent up to order n + 1 is
¿1,1 0 0 0
A2,i 0 0 0
, 0 —¿1,1 0
¿4,1 0 — 2^2,! —2Ai,i
¿n—2,1 0 n-4 (VMn-M —2 (n-2\ A n-5 V 4 5,1
¿n—1,1 0 n-3 ("3 ) An—3,i —2 (n-1\ * n-4l 4 ) 4,1
An,i 0 nh (n) An-2,l (4) An~ 3,1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
2 A
- |A3,1
- 3fA4,i
- 5 A4,1
■(n—4) 1 (n—2) ¿1,1 0 0 __ n— 1
■(n—4) 2 (ra—D ¿2,1 — (n—3) (n— A A 1 (n— JAM 0 2 A , nAn,1
■(n—4) 3 C 2) ¿3,1 — (n—3) / n A A 2 L—JA2,1 — (n—2) (n\ A 1 Uai,) Bn
0 0 0 2 ¿1,1
With these coefficients,
0 = ^n+1
0 — ^ind
u + F.
Then we apply the derivation £(•) to the hypersurface equation (20,5) written up to order n + 2, and we are going to compute ^L (—u + F(x)) ^, namely we are going to find <\d2W Of
0
I'2 l'y*2 ry sy
"21 + 2T" + ' ' ' + + Fra+2'0'-'0(nT2)i
,77 '1 '2 j-, '1 'n
+ ^n+1,1,...,0T-. -i \ i + ■ ■ ■ + ^n+1,0,...,1
(n +1)!
(n + 1)!
+ I A1,1'1 - 2¿2,1
' 1 "21
ry2 ry ryn ry
+ .Ay 1^2 + + 1 n
' 1 n
+
2
x1x2
IT
++
2!
'n—^
(n - 1)!
n
)
+ ^n+2,0,...,0
'
,n+1
(n + 1)!
)
+ Fn+1,1,...,0^--+ ' ' ' + Fn+1,0,...,1
n! n
/y»n ry \
x 1 x n 1
~nr J
+ I A2,1'1 - ^¿3,1
' 1 L2Î
2
(n - 1)! J
' 1 ¥
+
2
' 1 ' 2 Hi
+ ■ ■ ■ +
X^X2 'n 'n-1
+ ~2T + " ' + 'n-1 '
1! (n - 2)!
)
+ ^n+1,1,...,i
'
,n+1
+
(n +1)! 1!(n - 1)!
/V»n ry \
x n \
(n-J J
2
+ I A3,1'1 - *A1,1'3 - 4¿4,1
' 1 L2"
2
2 n—2
'1 'n—2
+ ^T + "' + (n - 2)! J
' 1 31
+
3
ry ' J ry _ ' 1 ' 2
2ÏÏ1
+ ■ ■ ■ +
'n-1'
^n—2
2! (n - 3)!
+ Fn+1,0,1,...,l
'
,n+1
+
/V»n ry
^n—1
(n +1)! 2!(n - 2)!
)
+
+ ¿^l'l - *Ai—2,1'3-----*¿2,lX¿—1 - (ï - 2)Al,l'i - ——¿i+1,1
2 1
L2"
+
2
' 1 ' 2 ~2T
++
'
n—i+1 1
'n—i+1
( n - + 1)!
)
1
1
(
Jb 1
+
x\x2
(i — 1)!1!
+ ■ ■ ■ +
Xi Xn—i+1
(i — 1)!(ra — ¿)!
+ ^n+^o-i-o
~n+1 i
+
Xn—i+2
(n +1)! (i — 1)!(n — i + 1)!
)
+
+ An-3,1^1 — *An-5,1X3-----*^2,1xn-4 — (n — 5)Ai,iXn-
2
n — 2
„n— 3
-A
n-2,1
X 1
L2Ï
234 + X1X2 ++ X1X3 ++ X1X4
2!
X1
+
Xn~3X2
+
3!
xI~2x3
+
4!
X1 X4
(n — 3)! (n — 4)!1! (n — 4)!2! (n — 4)!3!
+ ^n+1,o,...,1,o,o,o
X
n+1
+
/yn /y
i5
(n +1)! (n — 4)!4!
!4!^
+ An-2,1X1 — *An-4,1X3-----*^2,ixn-3 — (n — 4)Ai,iXn-2
n— 1
An-1,1
X1
L2Ï
23
x ^ x 3
2!
3!
)
X
n-2
+
xi X2
+
xi X3
(n — 2)! (n — 3)!1! (n — 3)!2!
+ ^n+1,o,...,1,o,o
X
,n+1
+
/y»n rf
i4
(n + 1)! (n — 3)!3!
)
+
2
An- 1,1X1 — *An-3,1X3-----*^2,1xn- 2 — (n — 3)Ai>iXn-1--An,1
n
X1
L21"
+
2
~2T
x
n— 1
+
x, X2
+ ^n+1,o,...,1,o"
X
.n+1
+
,-y»n /y _
X1 X3
!2!
)
+
(n — 1)! (n — 2)!1! n+1,0,...,1,0 (n +1)! (n — 2)!2!
An,1X1 — *An-2,1X3-----*^2,i Xn-1 — (n — 2)Ai,iXn + Bn
/y 2 -I i
L2T.
rf>n i
n!
ry<n ^n+1 ry*n ry*
x2 jp ^2
+ --TVTTT + ^n+1,o,...,o,1"-. - M +
(n — 1)!1!
(n +1)! (n — 1)!2!
)
)
By careful inspection of the products and after relevant simplifications, we find exactly the same factors (up to sign) of e1 as in the equations preceding Lemma 21.1:
0
(n
n+2 ( Ï
+ 2! < n Fn+2,o,...,o ¿1,1 + E (n + 2) Fn+i,o-1-o ¿fc,1 + ("J2) Bn >
)! I 2<fc<n J
+ Ö {<" — » F»+i,i.....o M
Xn+1X3 I (n +1)! I
-2 + 3
+ , 1 , -, :.< (n — 2) Fn+i,o,i,...,o ¿1,1 + * ¿2,1 + ■ ■ ■ + * An-2,1 + 7-(n +1)! An-1,1
(n + 1)! (n — 1)!3!
Î
+
+ f i . Xk,{ (n — k + 1) Fn+i,o-i-o ¿1,1 + * ¿2,1 +-----+ * A n—k+1,1
(n + 1)!
2 + k , .. . + (n — k + 2)!fc! + 1)! A^fc+2,1
}
3
i
2
i
i
i
i
+............................................................
+ fw {2 F"+1,».....1,» ¿1,1 + * ¿2,1 + l2^ (n + 1)! ¿3,1
+ TOT 1 F"+1,0.....0,1 ¿1,1 + -2+T (n + 1)! ¿2,1 y
In conclusion, this computation is a shortcut of the longer computation done previously with Examples 22.1. In dimension n = 2,-
41 {2 ^4,0 ¿1,1 + 4 F31 ¿2,1 + 6B2},
+ ^{^ia),)} ■
n = 3
'5 5!
I {3 F5,0,0 ¿1,1 + 5 -4,1,0 ¿2,1 + 5 -4,0,1 ¿3,1 + 10 B3},
4!
n = 4
„6
+ 2 F410 ¿1,1 j,
4
^ {-4,0,1 ¿1,1 + 2 ¿2,1 j■
' 6!
1 Ï 4 ^6,0,0,0 ¿1,1 + 6 F5,1,0,0 ¿2,1 + 6 F5,0,1,0 ¿3,1 + 6 ^5,0,0,1 ¿4,1 + 15B4},
5
+ X5XX2{ 3^5,1,0,0 ¿1,l}
5!
x)XW 10 ^
\ 2 F5,0,1,0 ¿ ,1 - 2 F5,0,0,1 ¿2,1 + — ¿3,1 j ,
.1 i — ^5,0,1,0^^1,1 —^ 5,0,0,1 ^^2,1 1 0
5! 3
5
5!
n = 5
fl^l 1 -5,0,0,1 ¿1,1 + 5 ¿2,)} ■
XTl1 |5 F7,0,0,0,0 ¿1,1 + 7 -6,l,0,0,0 ¿2,1 + 7 F6,0,1,0,0 ¿3,1 + 7 F6,0,0,1,0 ¿4,1 + 7 F6,0,0,0,1 ¿5,1 + 21 B5 |,
4 - 6,1,0,0,0 ¿1,1 ,
'
7!
x) x2
+ "6T
' 16 ' 3 10
+ 6! |3 —6,0,1,0,0 ¿1,1 - 2 —6,0,0,1,0 ¿2,1--3 Ft6,0,0,0,1 ¿3,1 + 5 ¿4,1 J ,
' 6 ' 4
+ { 2 F6,0,0,1,0 ¿1 ,1 - 5 F6,0,0,0,1 ¿2,1 + 10 ¿3,1 >,
6
14
\--— i 1 -6,0,0,0,1 ¿1,1 + 9 ¿2,1
1-6,0,0,0,1 ¿1,1 + 9 ¿2,11 ■
6!
n = 6
„8
X1 f
"8! |6 F8,0,0,0,0,0 ¿1,1 + 8 F7,1,0,0,0,0 ¿2,1 + 8 F7,0,1,0,0,0 ¿3,1 + 8 F7,0,0,1,0,0 ¿4,1 + 8 -7,0,0,0,1,0 ¿5 ,1 +8 -7,0,0,0,0,1 ¿6,1 + 28 B61,
+
7
X\X2
+ {5 F7,1,0,0,0,0 ¿1,11,
7!
x[XW 10
{4 Fj,0,1,0,0,0 ¿1,1 - 2 F7 ,0,0,1,0,0 ¿2,1 -—F7 ,0,0,0,1,0 ¿3,1 - 5 Fj ,0,0,0,0,1 ¿4,1 + 7 ¿5,1 k
x^X4 \ 35
| 3 F7,0,0,1,0,0 ¿1,1 - 5 F7 ,0,0,0,1,0 ¿2,1 - 10 Fj ,0,0,0,0,1 ¿3,1 + — ¿4,1
„7 „
,
+--7x^{ 2 F7,0,0,0,1,0 ¿1,1 - 9 F7,0,0,0,0,1 ¿2,1 + 21 ¿3,11,
7
+X1Xl {1 F7,0,0,0,0,1 ¿1,1 + 14 ¿2,11.
7!
23. Normalizations at order (n + 2) via jet prolongations
As before, we treat u = u(xi,...,xn) as a function of (xi,..., xn), The letter F will not be used. Then to each partial derivative ux^i,„^n (x1,..., xn), we can associate an independent coordinate (variable), denoted similarly or sometimes more simply Background appears e.g. in [11].
For each integer k ^ 0 we introduce a jet space of order k, namelv, R +( « ) equipped with the coordinates
(xl,...,xn,U, (u.^^nn) ).
V V 1 J -----h^n^K /
We shall employ an abbreviation:
u(k) := (ux-i x-n
1 n 1 • • •
For i = 1,... ,n we also introduce also total differentiation operators
O sr-^ ST^ O
:= ox + U- OU + Z Z
dxi xi du ^ ^ xixi •••x" du»i
m=1 +-----+ in=m xl • "x™
which commute one with another.
Given a general vector field in the (x, u)-spaee
n O O
L = Z Mx,u) ox. + u(x,u) dU>
j=1 1
its extension [11] to the infinite jet space
= L + ^ Uvi...
, , ^ OU
h-----X1 Xn
expresses how the (differentiated) flow of L acts on higher order jets, and its coefficients U^,...,, are uniquely determined by the formulae
(U - E X ^xi) + E
\ 1 <i n <i <n ' 1 <i n <i nn
Uvi,...,vn := Dx-1 ■ ■ ■ Dx-n U - N Xi ■ uxJ + > y ■ uxix':i-x7i.
^n X1 ^n \ / J " I / J • ••■Jjn
It is known that they depend on jet coordinates of order not exceeding v1 + ■ ■ ■ + un
U = U (t U U(l/1+"hn)N
According to Section 19 (see also the matrix in the beginning of Section 22), the general vector field, which stabilizes the normalizations up order n +1, i.e., which is tangent up to
n
order n + 1 to (19,1), reads as
r ^ 2 . \ d 2 . \ d . 2 . \ d
L = ¿1,1^1 - -¿2,1 u ---+ ¿2,1X1 -77¿3,1« T--+ ¿3,1^1 - ¿1,1 %3 - T¿4,1«
2 , j dx1 V , 3 , J dx2 V , , 4 , ) dx3
+ | An_ 1,1X1 - ^^(n31)An_3,1X3-----^-y-31 - ^¿„,1
n— 1
+ ( ¿ra,1x1--^r (3) ¿n-2,1^3-----(J!]J ¿2,1xra— 1 - 2 (^¿1,1X„ + BnU d
\ JL
J dxn
n\3l n~ 2,1^3 p. 1 1 -j VW rn"> I n
n - 2 V3/ 2 yn 1 1 vra/ I oxn
9
+ 2AUU—.
ou
Then we extend it to the jet space of order n + 2:
L(n+2) = L + £ Uvi..,^ (X)% + ... + £ (X)% - ^
U
^iH-----h^n = 1 ^iH-----+vn=n+1
du
Vi,,,,,V.
n
=n+2
'n
The following (admitted) statement can be established in a general theoretical context.
Lemma 23.1. If L is tangent to (20.5) up to order n + 1, then at the origin (x,u) = (0, 0), it holds:
0 = Uvi,,,,tvn (0, 0, u(vi+"+n)) for all U1 + ■ ■ ■ + un ^ n + 1.
Furthermore, at order equal to n + 2, by considering only independent jets, still at the origin, it can be shown in a general theoretical context that one recovers, up to sign and a change of notation, the expressions appearing in Section 22,
Lemma 23.2. If L is tangent to (20.5) up to order n + 1, then at the origin (x,u) = (0, 0), the identities hold:
Un+2,o,,,,,o (0, 0, u(n+2)) = - nun+2,o,...,o ¿1,1 - E (n + 2) «re+1,0—1—0 ¿fc,1 - (™+2) B, Un+1,1,,,,,0 (0, 0,u(n+2)) = - (n - 1) «n+1,1,,,,,0 ¿1,1,
Un+1,o,1,,,,,o (0, 0,«(ra+2)) = - (n - 2) un+1,o,1,,,,,o ¿1,1 - * ¿2,1-----* 2,1
-2 + 3
(n - 1)!3!
(n + 1)! An_ 1,1
Un+1,o-1-o (0, 0, u(n+2)) = - (n - k + 1) wra+1,o^1^o ¿1,1 - * ¿2,1-----* ¿n-k+1,1
-2 + k
(n - k + 2)!fc!
(n + 1)! ¿^k+2,1
^n+1,o,,,,,1,o (0, 0, u(n+2)) = - 2 Fra+1,o,,,,,1,o ¿1,1 - * ¿2,1 - + n -1 (n +1)! ¿3,1 v ' 3!(n - 1)!
—2 + n
Un+1,o,,,,,o,1 (0, 0,u(n+2)) = - 1 Fra+1,o,,,,,o,1 ¿1,1 --2,+n (n +1)! ^2,1.
Example 23.1. For instance, as n = 3, the vector field from Section 17 reads as
r {A 2A ^ & {A 2. \ &
L =1 ¿1,1 X1 - ^¿2,1 Uj — + ( ¿2,1 X1 - 3^3,1 Uj —
f , „ 2, \ 9 . 9
+ ( ¿3,1 X1 - ¿1,1 X3 - 4 ¿4,1 Uj — + 2 ¿1,1 u—, has continuation of order 5 above the origin (x,u) = (0, 0) given by
9
L(5) = - (3 «5,0,0 ¿1,1 +5 «4,1,0 ¿2,1 +5 «4,0,1 ¿3,1 + 10 B3)
9u
5,0,0
9
- 2 U410 ¿1,1 ^--(u4,o,1 ¿1,1 + 2 ¿2,1) ^-.
9U4,1,0 9U4,0,1
Hence, for n = 3, in the space R3 g (u5,0,0, u4,1,0, u4,0,1) of pure (n + 2)-jets, we have a linear space of vector fields parametrized by four free coefficients ¿\y1, ¿2,1; ¿3,u B3. This is in fact the Lie algebra of the action of the stability group in orders ^ n + 1 acting on pure n + 2 order monomials.
Without computing the flows of these vector fields (which would amount to recover the formulae of Section 15), we can realize that the two normalizations
—4,0,1 := 0 —5,0,0 := 0
are possible.
• By taking ¿2,1 := 1 and the others zero, we have
9 9
L(5) = -5u4i0^--0---2
9u5,o,0 9u4,1,o 9u4,0,1 and the flow of the constant vector field along the U4i0i1-axis clearly crosses the axis {u4,0,1 = 0}. • Assuming u4,0,1 = 0, by taking B3 := 1 and the others zero, we have
l(5) = _10 * _0^__0 9
9u5,0,0 9u4,I,0 9u4,0,I ' a vector field that stabilizes |u4,o,i = 0} and whose flow crosses the axis {u5,0,0 = 0}.
24. Normalizations of order (n + 3) monomials
Thanks to Proposition 21.1, several order n + 2 independent monomials can be normalized to zero:
—ra+2,0,...,0 := 0 —ra+1,0,1,...,0 := ° ...... —ra+1,0,...,1 := ° (24.1)
n + 3
as before the body-dependent ones, we get the equation
2 2 ra m 1 If rï~' rï~' r\ / ry ' ' ° ry I ry , ry
1 1 2 \ ( 1 m m_1 ^ A x
u = —1 +—^--+ x 11 1 1 x '
(ry 1 1 V ry I ry ry \
■^1 ^m m—1 \ ^ ^ \
+ ^ 2 (i - 1)1(7 - 1)!)
m=3 ¿,j>2 ^ ' '
i+j=m + l
X^X, +0 + ■■■ + 0 + X« 1
+ 0 +-n+1,1,...,0 t-—) + 0 + ... + 0 + X™ J]
(n +1)1 1 fa 2 (i - 1)1(j - 1)1
i+j=n+2
Xn+2X Xn+2X„ Xn+2Xo Xn+2X
X1 X1 X1 X2 J-, X1 X3 J-, X1 Xn
+ -n+3,0,...,0 7—7^77 + -n+2,1,...,0 7——7777 + -n+2,0,1,...,^ 7——7777 +----+ ^n+2,0,...,1 7—7^:77
(n + 3)1 (n + 2)1 (n + 2)1 (n + 2)1
+ «.+".....o + :.•;+' £ ^TTn^TDT + (3) + o.(n + 4),
n\ 1 2 (i - 1)!(, - 1)!
i+j=" + 3
where the border-dependent monomials written in the last line involve a supplementary monomial Fn+1,1,,,,,o ; this can be confirmed by reasoning as in Proposition 12,1,
We come back to the expression of a general affine vector field L tangent up to order n + 1, Taking into consideration current normalization (24,1), by looking at the equation in the end of Section 22, we see that the tangenev of L up to order n + 2 requires
¿2,1 := 0, ¿3,1 := 0, ......, An_ 1,1 := 0, Bn := 0,
hence coming back to (20,1), we get the following vector field:
L =¿1,1X1, dXl +0 5^ -¿1,1X3^3 +------(n - 3)^1,1^-1 - ®xn-i
+ (¿n,1^1 - (n - 2^Mxra) dxn + 2 ¿1,1 u du,
the flow of it, according to Lemma 20,1, at the next order stabilizes monomials of order not exceeding n + 2,
In order to see the action on order n + 3 monomials, we apply ^+3(0 to the tangenev equation, and not writing terms which do not contribute, we find:
Xra+2xi x"+2x2
0 = - 2 ¿1,1 ■■■ + ^n+3,0,,,,,0 7-, „1, + ^ra+2,1,,,,,0
(n + 3)! ' ^+2,1,,,,,0 (n + 2)!
1 + Xn \ ^+2)lj
,77 X1 x3 X1 xra
+ ^ra+2,0,1,,,,,^7-T7T\7 + ■ ■ ■ + ^ra+2,0,,,,,1
/ xn+2
+ ¿1,1 xA ■■■ + Fn+3,o,...,o r , + F, V (n + 2)!
(n + 2)! ' -+2,0,,,,,1 (n + 2)!
X^ ^ X^ X2
(n + 2)! ' ~"+2,1,,,,,° (n +1)!
,7? X1 x3 j-, ^n
+ ^ra+2,0,1,,,,,07-—rrr + ■ ■ ■ + ^ra+2,0,,,,,17-: T77
(n +1)! (n + 2)!
A / | 'X1 \
+ 0 - A1,1X\ ■■■ + F™+2,0,1,,,,,0 (^+2) + 2!(n - 1)!)
- 2 A1^{ ■■■ + F™+2,0,0,1,,,,,0 (^ + ^ 3!(^-12)!) +.............................................
- - 4MMxra_^■ ■ ■ + i?ra+2,0,,,,,1,0,0+ ^ + ( - (n - 3) Alt1X^ 1 - ^,1 [§ | + ^ + ^])
,n— 1 1
X1 X1X4
(n - 1)! + 1!(n - 2)! + 2!(n - 2)! + ra+2,0,,,,,1,0 (n + 2)! + 3!(n - 2)!
/. , \/Xi X1 X 2 . Xi XiX3\
+ (A.,1,1 - (n - 2MU*.) ++ f„+2,0.....1 (--+-2)! + j
We then collect all the independent monomials of order n + 3:
0 =X™+3< - 7-—rTFn+3,o,,,,,0A1,1 + 7-^+3,0,^^1,1 + -Fn+2,0,,,,,1An,1 \
{ (n + 3)! (n + 2)! (n + 2)! J
21
+ Xnl+2X2< 7-^77^n+2 ,1,,,,,0A1,1
1 (n + 2)! (n + 1)!
+ Xn+2X3^ t , Fn+2,0,1,...,0¿1,1 + , 1^llFn+2,0,l,...,0¿l,l
/ , o^,Fn+2,0,1,...,0¿1,1 + 7-"-.s,
(n + 2)1 (n + 1)1
1 JP A 2 1
T-"T:T7Fn+2,0,1,...^1,1--7—-777 ¿n,1
( n + 2)1 n 31( n - 1)1
2 1 1 1
г¿n,1 + 7777-777 ¿n,1
n 21 (n - 2)121 21(n - 1)1'
+ 7-7^7Fn+2,0,0,1,...^1,1 + 7-. -, M Fn+2,0,0,1,...,0¿1,1 - 7-. 0M Fn+2,0,0,1,...,0¿1,1 (
[ (n + 2)1 (n + 1)1 (n + 2)1 J +.................................................................................
2 f -2 1 + x. Xn-^S 7-, . Fn+2,0,...,1,0^1,1 + -7-777Fn+2,0,...,1,0,0¿1,1
[(n + 2)1 (n +1)1
n - 4 F 4
f Fn+2,0,..., 1,0,0 ¿1,1
(n + 2)!"
+ x™+2xra_i< 7-, Fn+2,0,...,1,0^1,1 + 7-, n+2,0,...,1,0^1,1 — -—37 Fra+2,0,...,1,0^1,1 (
[(n + 2)! (n + 1)! (n + 2)! J
+ xi xn S ~ ; ZT7Fn+2,0,...,1^1,1
{(n + 2)! (n + 1)! (n + 2)! J
After simplifications, this becomes
0 =xn+3{ (-T^TFn+3,0,...,0¿1,1 + (-T2)TFn+2,0,...,1¿n,1}
(n . /
,«+0. n
(n + 2)1" n - 1
1 ' 7-, o ^ I ^^ n^^2,0, 1 ,...,0 1, 1 + 77—¡"¿n,1
+ x"+2x^ (n + 2)1 Fn+2,1,...,0¿1,l}
+ x«+2x^ , o;,Fn+2,0,l,...,0¿l,l + — A n + 2
n 2
(n + 2)1 n+2,0,1,...,--,^ 31nr
~~+ X1 X4 T-rTTFn+2 0 0 ,1,...,0 ¿1,1
l(n + 2)1 +...............................
4
(n +2)1"
a Xn-1i 7- 0s . Fn+2,0,...,1,0^,
+ xT«+2Xn-2^ J^^—:-{F'n+2,0,...,1,0,0¿1,l^
+ xn+2xn-^^ 7-3 777Fn+2,0.....^¿1,11
(n + 2)1" 2
+ xi+2x^ (n + 2)!Fn+2,0,...,1^1,^.
Hence, we can normalize as
Fn+2,0,1,...,0 := 0,
n + 3
( n + 3)
2 2 n m 1
ry2 ry2 ry / ry ' '0 ry I ry ry
U =2+2 + +Xl ^ 2(i - 1)1(j - 1)1.
m=3 ¿,j>2 ^ ' '
i+j = m + 1
+ Fn+1,1.....0(n+1XT + 0 + - +0 + XÎ £ 1 ""
(n +1) 1 ^ 2 (i - 1)l(, - 1)1
i+j=n+2
+ F
n+3,0,...,0
X1 X1 , J7 X1 X2 I n I Z7 X1 X4
+ ^ra+2,1,...,0 -"TÄT + 0 + ^n+2,0,0,1,...,0 "-~ +
+ Fr
n+2,0,...,1
{n + 3)! {n + 2)!
+ Fr
{n + 2)!
+ X2X2
{n + 2)!
n+1,1,...,0
m
+^ E
1
'X ß 'X j
i,3>2
i+j=n+3
2 {* - 1)!{j - 1)!
+ Ox/ (3) + Ox(rc + 4), we deduce that
Ana := 0.
The isotropy Lie algebra is 1-dimensional and is represented by the matrix:
¿1,1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 -¿1,1 0 0 0 0
0 0 0 -2AM •• 0 0 0
0 0 0 0 ■ -{n - 3)^1,1 0 0
0 0 0 0 0 -{n - 2)Ä1,1 0
0 0 0 0 0 0 2 ¿1,1
25. Orders (n + 4) and (n + 3) The corresponding matrix Lie group
«1,1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 «1,1 0 0 0 0
0 0 0 1 . «2,1 ■ 0 0 0
0 0 0 0 1 n — 3 «i,i 0 0
0 0 0 0 0 1 n-2 «1,1 0
0 0 0 0 0 0 <1-
consists of the plain dilations yi = ai,i xi, y2 = 0,
V3
«1,1
■X3,
Vn
n-2 X'n 1,1
Lemma 25.1. AH power series coefficients Fc
ai,...,an
are relative invariants.
a2 1 u.
Proof. Possible remaining maps fixing the origin which send a hvpersurface normalized as above
u =
£
ai,...,a„
rfPl . . . rf-in p
1 i/y,^ J- (
n A ai,...,an
to a similarly normalized hvpersurface
v = £ V? ••• Van
o
are only the dilations
yi = a xi, y2 = 0,
ai,...,an
1
y3 = - %3, a
V'n an-2 Xn,>
v = au.
1
1
v
1
where we have abbreviated a1,1 =: a. After replacement
1 \ °3 / 1 \
a2u = Z (axi)ai (x2)a2 (l^ "' (o-ä^)
a1
an identification gives:
—ai,...,an a a a ' ' ' a ( ) GCTi,...,(jn.
Thus, each pair of power series coefficients are nonzero multiple one of the other. Thus, they vanish (or do not vanish) simultaneously, □
Consequently, at the next two orders n + 4 and n + 5 we shall not perform any further normalization since this would create some branching. But we must determine the independent and border-dependent monomials of homogeneous degrees n + 4 and n + 5,
Proceeding as in Lemma 12,1 to take account the assumption that the Hessian matrix is of 1
computations.
Theorem 25.1. In each dimension n ^ 2, every local hypersurface Hn C Rn+1 having constant Hessian rank 1 which is not affinely equivalent to a product of Rm (1 ^ m ^ n) with, a hypersurface Hn-m C Rn-m+1 can he affinely normalized as
ry2 ry / ry^^ ry 1 ry ry \ ryn I1
X1 x1x2 / X1 xm ™-1 V^ 1 Xl'X3 1 1 J7 X1
U = T + " + ¿3,1" +X' ¿2 2« - 1)!(i- -+"1)!
i+j = m + 1
I „1 V^ 1 Xi Xj__L 77 Xl+| 77 Xl+ X2
+ X1 Ö r TTTT7 rw + —1+3,0-0 7 ——7 + —1+2,10-0 7 —777
2(% - 1)!0 - 1)! (n +3)i (n +2)i
i+j=n+2
^^l^ ry ry
X1 X4 X1 Xra X1 x2x2
+ —1+2,0010-0 7-77777 + ' ' ' + —1+2,0-01 7-7—77 + —1+1,10-0 -j-
(n + 2)! (n + 2)! ni
._, 1 X X X1+4 X"+3 X
I „1+1 V^ 1 Xi Xj__1 77 X1__I 77 X1 X2
+ Xl fa 2 (i - 1)!(j - 1)! + Fl+4'°-° (^+4)! + Fl+3'10-0 (^+3)
i+j=n + 3
X1+3 X Xl13 X Xl13 X
X1 X3 X1 X4 X1 Xl
+ —1+3,010-0 7-7—77 + —1+3,0010—0 7-7—77 + ■ ■ ■ + —1+3,0-01 7-7—77
(n + 3)! (n + 3)! (n + 3)!
—1+2,10-0 —1+1,10-0 —1+2,0010-0 —n+2,0-01
—- . X2 X2 +--—-j-X2 X3 +--}-: 777 X2 X4 + ■ ■ ■ + —-7—TT X2 X
(n + 1)! 2! n! (n + 1)! (n + 1)!
+ ^ ^ 1 Xi Xj
+ X1+2
1
^2 2 ( - W - 1)!
i+j=n+4
X1+5 X1+4 X X1+4 X
X1 X1 X2 X1 X3
+ —1+5,0-0 7—7—77 + —1+4,10-0 7—7—77 + —1+4,010-0 7—7—77
(n + 5)! ( n + 4)! (n + 4)!
X1+4 X X1+4 X
X1 X4 X1 X1
+ —1+4,0010-0 7—7—77 + ■ ■ ■ + —1+4,0-01
J- ■ -0 / | I I ^ 1+4,0-01 /
(n + 4)! (n + 4)!
+ X11+3
—1+3,10-0 —1+3,0-0 \ + I —1+3,010-0 + —1+2,10-0 \ (n + 2)! - 2! (n +1)!yX2X2 + V (n + 2)! + 2! (n +1)^X2X3
1+3,0010 0 —1+1,10 0 —1+3,00010 0 —1+3,0 01
(n + 2)! + "Hn JX2X4 + (n + 2)! X2X5 + ■■■ + 1^+W
+ Fn+2,0010-0 + + Fn+2,0-01 + 1 XiXj
+ 2!(n + 2)! X3X4 + '' 2! (n + 2)!X3Xn + fa 2 (i - 1)!(j - 1)iJ
i+j=n+5
+ 0X2t...tXn (3) + 0XltX2t...tXn (n + 6). This explicit expression of the graphing function F(xi, ..., xn) is our new starting point, 26. Summary of proof of main Theorem 1.3
L — + C\ X1 + C2 X2 + ■ ■ ■ + Cn—1 Xn— 1 + Cn Xn^
We take a general affine vector field which does not necessarily vanish at the origin:
d_ du
+ (Ti + An Xi + A12 X2 + ■ ■ ■ + Ai ,n— 1 Xn— 1 + A1,n Xn ) r\
+ (T + A2,1 X1 + A2,2 X2 +-----+ A2,n—1 Xn—1 + A2,n X^
+
+ (Tn—1 + An—1,1 X1 + An—1,2 X2 + ■ ■ ■ + An—1,n—1 Xn— 1 + An—1,nXn^
Xn 1
+ ('Tn + An,1 X1 + An,2 X2 + ' ' ' + An,n— 1 Xn— 1 + An,n XnJ
dx,n
We recall that if L is tangent to the hvpersurface H — {u — F(x1,... ,xn )}, then T0 — 0, since u — F — 0X(2) Here the parameters T1,... ,Tn are tightly related to the infinitesimal
L
L10 — T1 n--+ ■ ■ ■ + Tn 7;-,
10 ax1 oxn
since homogeneity requires
T0H — SpanR {L|0: L|# tangent to Hj, and since, again due to F — 0X(2),
T0H — Span {7—, ..., 77— Iox1 oxnJ
T1 , . . . , Tn
■should remain absolutely free in all computations.
Such general afiine vector field L is an infinitesimal affine symmetry of our hvpersurface H — [u — F(x1,..., xn)} graphed as in Theorem 25.1 if and only if L|# is tangent to H and if and only if the following power series identity holds in R{x1,..., xn}:
0 = L (-u + F) .
u=F
We shall in fact 'only' study independent monomials of order not exceeding n + 4 in this fundamental equation, namely, we shall examine u + F)| F)- Recalling that
fxi — Ox(1), fx2 — Ox(2), ••• fx„_ i — 0X(n - 1), fx„ — OX(n),
we can therefore begin with writing
0 = A0 + A1 + ■ ■ ■ + An—1 + An,
where
A0 :— - C1 X1 - C2 X2-----Cn—1 Xn—1 -CnXn -D [<+4(F)],
A1 := ÎT1 + AMX1 + A1 2 X2 + ■ + A1,ra- -1 Xn- 1 + A1, n Xn + B1 [<+3 (f )
A2 := fe + A2,1 X1 + A2 2 X2 + ■ + A2,„- -1 Xn- 1 + A2,nXn + B2 [<+2 ( F )
A3 := (T3 + A3,1 X1 + A3 2 X2 + ■ + A3,„- 1 X n- 1 + A3, n Xn + B3 № ( F )
A4 := T4 + A4,1 X1 + A4 2 X2 + ■ + A4,n- 1 X n- 1 + A4, n Xn + B4 K d( F )]
A5 := T5 + A5,1 X1 + A5 2 X2 + ■ + A5,n- 1 X n- 1 + A5, n Xn + B5 fe1 ( F )
A6 := T6 + A6,1 X1 + A6 2 X2 + ■ + A6,n- 1 X n- 1 + A6, n Xn + B6 [<d2 ( F )
]) Fx
]) Fx
F,
X4 i
Of,
])f,
— 1Xn — 1 + An-l,nXn + Bn—1 Kd( f)]) FXn — 1 ,
An := ( Tn + An,1 Xi + An,2 x2 + ■ ■ ■ + An,n—1 Xn— 1 + An,n Xn + Bn
In the next Section 27, we shall compute some of the coefficients of the monomials in this large equation, namely, we shall find
E[ai,...,an] := [X<J1 ■ ■ ■ xann] (L (- u + F) Y ax + ■ ■ ■ + °n < n + 4,
V u=F/
which are linear in C,, D, T,, A,,^ B,, and which should vanish for L to be tangent to H:
E[ai,...,an] = 0.
In particular, we shall find
I := E[n+2,0,...,0,1] = °
II := El
[n+3,0,...,0,1]
0.
But before proceeding to the (non-straightforward) computations, let us summarize the key reason why affinelv homogeneous models do not exist in dimension n ^ 5. We use * to denote any unspecified real number whose value does not matter.
Lemma 26.1. For a hypersurface {u = F(x)} normalized as in Theorem 25.1, after taking into consideration -some of the other equations E[ai,. ,an] = 0, these two specific equations I, II become
^ ' ^ T4 + "-7777 Fn+2,0-01 A1,b
0 = * T1 + * T2 -
12 (n - 3)n! 4 (n + 2)!
1 3 0 = * Ti + * T2 + * T3 - —-— T5 +
30 (n - 4) n! 5 (n + 3)!
F
ra+3,0-01 Ai,i.
Admitting temporarily this fact, we can easily complete our main non-existence result.
Proof of Theorem 1.3. If the power series coefficient Fn+2,0_01 = 0 would be zero, then the first equation:
i „ 1
0 = * T1 + * T2 -
T4
12 (n - 3) n!
would consist of a nontrivial linear dependence relation between T1,... ,Tn, contradicting infinitesimal transitivity. Hence, Fn+2,0_01 = 0, But then we can find the isotropv parameter by I
A1,1 = *T1 + *T2 + *T4, that we replace in II, getting, whatever the value of Fn+3,0_01 is
0 = *T1 + *T2 + *T3 + * T4 - 1 T5.
30 ( n - 4) n!
But such an equation is also always a nontrivial linear dependence relation between T1,... ,Tn contradicting again infinitesimal transitivity, □
Observe that n ^ 5 was used in this argumentation,
27. Tangency equations at orders ^ n + 4 It remains to prove Lemma 26,1,
Proof of Lemma 26.1. We find A0:
An
- Cl xi - • • • - Cn Xn - D
2 2 n 1 n n+2
2 2 - n
1 1 2 1 n- 1 1 n 1 2 +--21--'----+ ~-^ +----Fn+1,10-0
( n - 1)1 n1
0 • • • 0 (n +1)1
+ Fn
X
n+3
n+3,0 0
• • • 0 (n + 3)1
+ Fn
n+2,10 0
X1n+2 X2
• • • 0 ( n + 2)1
+ Fn
n+2,0010 0
X1n+2 X4
• • 0 (n + 2)1
+ + Fn
n+2,0 01
Xn+2 X 1 n
• • ^ (n + 2)1
x« + Xn+ X2 „ X1 + X3 J-, X1+ X4
+ Fn+4,0-0 7-T7T77 + Fn+3,10-"0 7-7—77 + Fn+3,010-0 7-7—77 + Fn+3,0010-0
0 • " (n + 4)1
0 • • • 0 ( n + 3)1
0 • • • 0 (n + 3)1
0 • • • 0 (n + 3)1
+ ■ ■ ■ + Fi
n+3,0 01
Xn+3 X 1 n
• • • 01 (n + 3)1
Now we write A1; ,,,, An, which all involve products. We denote the products using the sign "•".Here is A1:
A1
(
tl + ¿1,1 xl + ¿1,2 x2 +-----+ ¿1,n-1 xn-1 + ¿1,n x.
+ B1
2 2 n
ry2 ry2 ry _ ry'° ry
Xl X^Xn
2Î + ~2T + " ' + ~
n1
n |1 |3
+ F X1 X2 + F X1 + Fn+l,l0-0 7—7—77 + Fn+3,0--0
0 • • • 0 (n +1)1
Xn+2 X Xn+2 X
+ F X1 X2 + F X1 X4 + + F + Fn+2,10-"0 7-7—77 + Fn+2,0010-0 7-7—77 + ' ' ' + Fn+2,0-01
• • • 0 (n + 3)1
Xn+2 X 1 n
(
0 • • (n + 2)1
0 0
(n + 2)1
• • • 01 (n + 2)1
2 n 1
2-
X1 X3 X1 Xn
xi + X1X2 + +-----+ 7-—7 + Fn+1,10-0
21
(n - 1)1
Xn X Xn+2
X1 X2 X1 -j--r Fn+3,0-0
n1
0 • • • 0 (n + 2)1
Xn|1x 1 n
+ F xi X2 + F xI X4 + + F _
+ Fn+2,10-"0 7-7— + Fn+2,0010-0 7-7— + ■ ■ ■ + Fn+2,0-"01 7-7—
( n + 1)1 ( n + 1)1 ( n + 1)1
+ Fn
X
n+4,0 0
• • • 0 (n + 3)1
n+3 X1n+2 X2 X1n+2 X3 X1n+2 X4 + Fn+3,10-"0 7-7—77 + Fn+3,010---0 7-7—77 + Fn+3,0010-0
0 • • " (n + 2)1
0 • • • 0 (n + 2)1
0 • • • 0 (n + 2)1
Xn+2 X 1 n
Xn+4 1
n+3 n+3
+ + F - 1 ~n + F + F X1 X2 + F X1 X3
+----+ Fn+3,0-01 7-7—77 + Fn+5,0-0 7-7—77 + Fn+4,10-0 7-7— + Fn+4,010-0 7-7—77
(n + 2)1 (n + 4)1 (n + 3)1 (n + 3)1
+ Fn
n+4,0010 0
X1n+3 X4
0 • • • 0 (n + 3)1 A2
+ ■ ■ ■ + Fi
n+4,0 01
Xn+3 X 1 n
• • • 01 (n + 3)
1.
A2
(
t2 + ¿2,1 xl + ¿2,2 x2 +-----+ ¿2,n— 1 Xn-1 + ¿2 , n X n
2
+ B
X
2
X21 X2
1 1 ^1^2 xlxn J-,
"2T + 2T" + ■■■ + + Fi
n+1,10 0
Xn|1X2
• • • 0 (n +1)1
rm-1r 1 m- 1
21 ' ^ 11 (m - 2)1
m=3 v '
X2 n 1 +E
\ m — 3
+ Fi
X
n+1
n+1,10 0
+
• • " (n +1)1 (n - 1)1
Xn X Xn+2
1 n 1
+ Fn
n+2,10-0
• • • 0 (n + 2)1
1
1
2 xra+3 2 Xra+2X3 + ^1+1,10-0 -T X x2 + ^1+3,10-0 -. o\, + ^1+2,10-0 -: 7TT + ^1+1,10-0 "7^-
n! (n + 3)! (n + 1)! 2! n!
X1+2X X1+2X x1+4
,77 X1 x4 j-, X1 X1 j-, X1
+ ^1+2,0010-0 7-7777 + ■ ■ ■ + p 1+2,0-01 -. 7V7 + ^1+4,10-0
+ ( ' i+3,0010v0 + ~ i+;,10^0) x^x^j.
(n + 1)! l+2,0•••0l (n + 1)! ' i+4,1°- (n + 1)!
2p1+3,10-0 p1+3,0-0\ ra+3 + /^1+3,010-0 + ^1+2,10-0 \ ra+
, (n + 2)! - (n + 1)! T1 X2 + V (n + 2)! + 2!(n + 1)!/X1
^1+3^0010-0 + ^1+1,10-0
(n + 2)! 3! n!
We find A3:
2 2 1
( XX X X2 \
A3 =(T3 + ¿3,1X1 + ¿3,2X2 + ■■■ +¿3,1-1 X„-1 + ¿3,„X„ + ^3 ^T + ^T + ■ ■ ■ +—)
K.3 J^ Xm—1X X1X X1+1X „.™+3
\ X1 xm_2 X1 xn t-!
+ 2^ 2! (™ _ 3!) + 2! (n - 2)! + 2! (n - 1)! + ^1+3,010^0
3 1
/ •X' 1 X A >Xj 1 >XJ mm—_2 X 1 X1_1 1 7—1 1
I 77 -1 ~3 I Z? __I I - 1+3,010-0 - 1+2,10-0 ! 1+3
+ p 1+1,10-0 "77—j--r ^1+4,010-0 7—7777 + ~t—TTTVi—+ öi~7—TTVi" X1 X2
3! ^2!(m - 3!) 2! (n - 2)! 2! (n - 1)! 1+3,01°^0 (n + 3)!
t ^1+3,010—0 + ^1+2,10-0 \
v (n + 2)! + 2!(n + 1)!J
x1+2x^ ^ F X11+4__L ( ^1+3,010-0 , p1+2,10-0 \ 1+3,-
2! n! n+4,010•••0 (n + 4)! \ (n + 2)! 2! (n +1)!,
X1+3 X X1+3 X
X1 X4 X1 X1
+ p 1+2,0010-0 777-. + ■ ■ ■ + ^1+2,0-01
2!(n + 1)! 1+2,°-01 2! (n + 1)!y
A4
A4 = + ¿4,1 X1 + ¿4,2 X2 +-----+ ¿4,1—1 X1—1 + ¿4,1 X1
+ ^4
(
2 2 1_1
X1 X1X2 X1 X1_1
2i + ~2T + ■ ■ ■ + (n - 1)!
X4 1 Xm_1X X1X X1+2 X1+1X
X1 \ ^ X1 xm_3 xlxn_2 j-! X1 X1 X1— 1
4i + 3!(m - 4)! + 3!(n - 3)! + 1+2,0010^0 (n + 2)! + 3! (n - 2)!
X1+3 X1+2 X X1+2 X
X1 X1 X2 X1 X1 + ^1+3,0010-0 7—. „m + ^1+2,0010-0 7—rm +
(n + 3)r n+2,0010•••0 (n +1)! 3!(n - 1)!
+ p x1+4 ^ ^1+3,0010-0 + F1+1,10-^ X1+3X + p x1+3x3 \
+ F1+4,0010-0 (nT4)i H (n + 2)! + JX1 X2 + F1+2,0010-0 2!(n + 1)! J'
Here is A5:
A5 ^(^5 + ¿5,1 X1 + ¿5,2 X2 +-----+ ¿5,1_1 X1_1 + ¿5,1 X1
+ ^5
2 2 i — 2 _
X1 + x1x2 + + X1 x1_2
2! 2! (n - 2)! _
x5 xm_1x x1+2 x1+1x ^
x A ^ ^ ^ 1—3 . ^ ^-'n_2
/ x5 _^
■ ( ßf + £ 4!1(m-5)! + 4!(^-_4|)! + F1+2,00010-0 (^+2)1 + 4! (n - 3)!
X1+3 X1+2 X X1+2 X
X1 X1 X2 X1 x1_1
+ p1+3,00010^^ 7 -777 + ^1+2,00010-0 7 —777 + 777 777
(n + 3)! (n + 1)! 4! (n - 2)!
X1+4 X1+3 X X1+3 X X1+3 X
X1 X1 X2 X1 X3 X1 X1
+ p 1+4,00010-0 7--777 + ^1+3,00010-0 7--7777 + ^1+2,00010—0 ^^——¡77 + 777-777
(n + 4)! (n + 2)! 2! (n + 1)! 4! (n - 1)!
We find A6:
T6 + ¿6,1 Xl + ¿6,2 X2 + ■ ■ ■ + ¿6 , n 1 X n 1 + ¿6 , n Xn
+ B6
(
X
X1 X^X2 xi Xn-3
2Î + 21 ■■■ + (n - 3)1
n Xm-1 Xm-5
61+s
51 (m - 6)1 51 (n - 5)1
n
1 n- 4
+ 7:77-^77 + Fn+2,000010---0
n+2 Xn+1 X 1 + Xl Xn— 3
• • " (n + 2)1 51 (n - 4)1
Xn+3 1
+ Fn+3,000010•••0 7 , I + Fn+2,000010•••0 / , -, M I K, / oM
( n + 3)! ( n+ 1)! 5!( n- 3)!
Xn+2 X Xn+2 X X1 X2 + Xl Xn—2
+ Fn
X
n+4
n+4,000010 0
• • " (n + 4)1
+ Fn
n+3,000010 0
X1n+3 X2
0 ( n + 2)!
+ Fn
n+2,000010 0
Xn + X3
+
Xn+3 X
n
• • " 21 (n +1)1 51 (n - 2)
)1!
A n 1
A
n 1
(
Tn— 1 + ¿n—1,1 Xl + ¿n—1,2 X2 + ■ ■ ■ + ¿ n— 1, n— 1 Xn—
1+ ¿
n— 1, n Xn
2 „2„ „,3r
+ Bn 1
(
2 2 3 4
X1 X1 X2 X1 X3 X1 X4
2 + + +
X
n1
+
2!
Xn— 1 X2
3!
+
4!
n
X X3
(n - 1)1 (n - 2)111 (n - 2)121
+ Fn
X
n+2
n+2,0 010
+
Xn+1 X4
(n + 2)1 (n - 2)131
+ Fn
X
n+3
n+3,0 010
( n+ 3)!
+ Fn
n+2,0 010
Xn+2 X2
+
Xn+2 X5
(n +1)1 (n - 2)141
n+3 n+3 n+3
1 X X2 X X3 X X6
+ Fn+4,0"010 7-T7T77 + Fn+3,0•••010 7-77777 + Fn+2,0-010 7777-. - M +
Xn+4 1
(n + 4)1
(n + 2)1
21 (n + 1)1 (n - 2)151
1)
A n
A n
(
Tn + ¿n,1 X1 + ¿n,2 X2 + ■ ■ ■ + ¿n,n—1 Xn—1 + ¿n,n Xn + Bn
(
2 2 3
rf' ^ rf' ^ rf> rf''j rf>
1 •X'^X'2 .X'^X'3
2!
3!
n
X1 X2
n
Jb 1 — +
n! (n - 1)111
+ Fn
X
n+2
n+2,0 01
+
Xn + X3
(n + 2)1 (n - 1)121
+ Fn
X
n+3
n+3,0 01
(n + 3)1
+F x1+2x2 + + Fn+2,0•••01 7-7-777 +
Xn X4
+ Fn
n+3,0 01
(n +1)1 (n - 1)131
Xn+3X2
(n + 2)1
+ Fn
X
n+4
n+4,0 01
+ Fn
Xn+3 X3
n+2,0 01
+
(n + 4)1 Xn+3 X5
21 (n +1)1 (n - 1)141
6
1
1
1
1
1
1
1
By looking carefully at these products, we can determine the coefficients of some relevant monomials.
First, to get equation I, we extract the coefficient of the monomial xn xn. By inserting vertical bars, we indicate from which A0 1... n the written terms come from:
Ac
- D
A2
—1
1+2,0-01 (n + 2)!
A
—1+3,0 01 —1+2,0 01 —1+3,0 0
+ — 1 —---r ¿1,1 —---r ¿1,-
+ T2 —+¿2,1
(n + 2)! —1+2,10 0
(n + 1)! 1
(n + 2)!
1
(n + 1)!
-^ + £2-:--+ £2-
(n + 2)! + 2 2! (n - 1)! + 2 2! n!
A3
+ ¿
1
3,1
A4
A6
+ ¿
6 , 1
2! (n - 1)!
—1+2,000010-0 (n + 2)!
Ay
1 + a —1+2,0010-0
3! (n - 1)! + ¿4,1> (n + 2)!
A5
+ ¿
—1
1+2,00010 0
5, 1
+
A„-l
+ ¿ 1
—1
1 1 , 1
1+2,0-010 (n + 2)!
+ ¿1
(n + 2)!
—1+2,0-01 (n + 2)! .
Second, to get equation II, we extract the coefficient of the monomial xn+3xn:
II
Ac
D
—1
1+3,0-01
(n + 3)!
A
—1
—1+4,0 01 —1+3,0 01 —1+4,0 0
+ —1 (n + 3)! (n + 2)! +¿1,1 "(^+3)!
+ R —1+2,0-01 + R —1+2,0-01 + 1 2! (n +1)! + 1 (n + 2)!
A2
—1+2,0 01 —1+2,10 0 + ¿2,1 —-TTVT + ¿2,1
+ . —1+3,010-0 + R + ¿3,1 —-, „m + £3
1
+ —5
(n + 3)! 1
+ #3
1
2! (n - 1)! 3 3! n!
A6
(n + 1)!
A4
+ ¿4,1
(n + 2)!
A3
—1
1+2,0 01
1
+ ¿
—1
1+3,00010 0
4! (n - 1)! ' (n + 3)!
An
+ ¿
—1
3! (n - 1)!
Ay
+ ¿
+ T3 —1+3,0010 0
2! (n + 1)!
A5
6, 1
1+3,000010-0 (n + 3)!
+
4, 1
A „_ 1
(n + 3)!
+ ¿ 1
—1
1 1 , 1
1+3,0-010 (n + 3)!
+ ¿1
—1+3,0-01 (n + 3)! '
We see that we need to determine the parameters:
¿1, 1, £1, ¿2,1
¿2, 1, £, ¿3,1
¿3, 1, £3, ¿4,1
in terms of T1, By
—15 ¿1,1-
¿1-2,1,
¿1—1,1, ¿ ,
1, 1
E [2,0,...,0] : 0 = 11 T2 + ¿1,1 - 2D,
we determine
0
0
D = T2 + 2 ¿1,1.
We then get:
E[1,0,...,0,1] : E[2,0,...,0,1] :
E[3,0,...,0,1] : E[n—2,0,...,0,1]
0 0
0
A1, n, 1
2j A2,n,
-A3
3! A3,n;
1
(n - 2)!
An
E[n—1,0,...,0,1] : 0 = T1
n— 2, n,
+
(n - 1)! (n - 1)! 1
An 1,
E[n,0,...,0,1]: 0 = D r + A1,1 7-+ T2 7-77 +--r An,n.
n! (n - 1)! (n - 1)! n!
Replacing the obtained values, we find:
n I (rp , o A \ Fn+2,0-01 . rp Fn+3,0---01 , Fn+2,0-01 , rp Fn+2,0-01 , D n 0 = - ( T2 + 2A1,1) —-7777 + T1 7-7777 + A1,1 —-7777 + T27-7777 + B2
+ A
3,1
(n + 2)! 1 (n + 2)! 1 +T4 1
(n + 1)!
(n + 1)!
2! n!
2! (n - 1)! 4 3! (n - 1)!
+ A Fn+2,0-010 + A Fn+2,0-01
+ An— 1,n / '. 7Ti + An
and:
(n + 3)!
01 + T1
(n + 2)!
Fn
(n + 2)!
„ II (rp 0 A X Fn+3,0-01 , m Fn+4,0-01 , A Fn+3,0-01 0 = - ( T 2 + 2 A1,1)
• • 0 1 A n+3,°-01 , R 77 777 + A1,1 7-7777 + B1 Fn+2,0-"01
(n + 3)! + 1,1 (n + 2)!
. , Fn+2,0-01 . rp Fn+2,0-01 . D 3 n + 2 . ,
+ A2,1 7-—77 + T 3 777-1 777 + B3 —-T + A4,1
1
(n +1)! 3 2! (n +1)!
+ a Fn+3,0-010 + a Fn+3,0-01
+ An— 1,n / \ 777 + An,n
12 n!
+ T5
n + 4 2! (n + 2)! 1
3! (n - 1)! 5 4! (n - 1)!
(n + 3)!
(n + 3)!
To find B2, B3, and A2,1; A3,1; A4,1; we consider by patiently chasing in A0, A1; the three equations:
A n
0
E[
[n+1,0,...,0,1]
Ai
rp Fn+2,0-01 D 1
T 17-7777 + B1
(n + 1)!
+ B 1 2! (n - 1)! + 1 n!
A2
+ A2,1
1
(n - 1)!
+ A
Fn
n+1,10-0
2, n
(n + 1)!
A3
+ T3
1
E[
[n+1,0,...,0,1,0]
Ai
T1
Fn
n+2,0-010
(n + 1)!
A2
+ A2, n 1
2! (n - 1)!'
Fn+1,10-0
+ B^-r-^——T + B2 1
(n + 1)!
A3
2! (n - 2)! 2 (n - 1)! 2!
+ A
E[
[n+1,0,...,0,1,0,0]:
Ai
T1
+ B3
Fn
n+2,0-0100 (n + 1)!
1
(n - 2)! 3!
A2
+ A2, n 2
Fn
n+1,10-0
° (n + 1)!
3,1
A3
2! (n - 2)!
A4
+ T4
3! (n - 2)!'
+ B3
1
A4
+ A
4,1
3! (n - 3)!
A5
+ T5
2! 2! (n - 3)! 1
4! (n - 3)!
0
1
1
0
1
1
0
1
in which the three underlined parameters vanish thanks to:
E[2,o,...,o,1] : 0 = ¿2,n 2,
e[2,o,...,o,1,o] : 0 = ¿2,n-1,
E[2,0,...,0,1,0,0] : 0 = ¿2,n-2 21!.
After simplification, these three equations become
0 = p —n+2,0-01 + p 1 + A 1 + B n + 2
0 = p inr^ + T3 2! (n - 1)! (n-^ + B1 ,
0 = T1 Fn+2,0:010 * n
(n + 1)! 4 3! (n - 2)! 3,1 2! (n - 2)! 2 2! (n - 1)!' 0 = —1 —+ T5 * ,+AM-, 3n - 4
(n +1)! 5 4! (n - 3)! 4,1 3! (n - 3)! 3 2! 3! (n - 2)!' 3 + 3 3
2 n
B1 = p F«,+2,0...01 (n +1)(n - 2) + p 3(n-2) ,
4 = P — 2 r 2 n - 1
¿2,1 = - P1 —n+2,0-01 (n +1)(n - 2) - P3 3 n-2,
and
£2 = T1 —1+2,0-010 7-774——TT + T4 1 -1,
( n - 3)n(n +1) 6 n - 3
4 = T F 6 T 1n - 2
¿3,1 = - T1 —1+2,0-010 (n - 3)n(n +1) - T4 2 n-3;
as well as
12 1 n- 2
B3 = p1 —«+2,0-0100 ^-77—-77- + p5 —--,
n(n - 4)(n2 - 1) 10 n - 4
24 2 n - 3
¿4,1 = - p1 —«+2,o-o1oo —f-77—-77- - p5 --7.
n(n - 4)(n2 - 1) 5 n - 4
I
0 == *T + *T +T (1 ^ «-2 1 . 1 \ 0 1 + 2 +4V 6 n - 3 2!n! 2 n - 3 2! (n - 1)! + 3! (n - 1)!y
AT? ( 2 1 n - 2 \
+ ¿M —1+2,0-0^- (n+2)T + (nTT)T- (n+2)T)
II
0 = *T, + ,T2 + *T3 + T.fi n-i 3n + 2 _ 2 n-j 1 1 \
5V 10 n - 4 12 ■n! 5 n - 4 3! (n - 1)! 4! (n - 1)!/
2 1 n - 2 \
+ ¿1,1 —n+3,0-0^ - + (n^ - J
which is also exactly what Proposition 26,1 stated, □
Acknowledgements
The author expresses grateful and warmest thanks to the Ufa Mathematical Journal, and to the Virtual Workshop "Complex Analysis and Geometry", organized by E, Garifullin, I. Musin, A, Sukhov, E, Yulmukhametov, Ufa, Eussia, November 16-19, 2021,
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Joël Merker,
Institut de Mathématique d'Orsay,
CNES, Université Paris-Saelav,
Faculté des Sciences,
91405 Orsay Cedex, France
E-mail: joel [email protected]
