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EDN: XIDJZF УДК 517.55
Series of Hypergeometric Type and Discriminants
Quang Khanh Phan*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.03.2023, received in revised form 15.04.2023, accepted 04.06.2023
Abstract. The monomial of solutions of a reduced system of algebraic equations are series of hypergeometric type. The Horn-Karpranov result for hypergeometric series is extended to the case of series of hypergeometric type.
Keywords: series of hypergeometric type, logarithmic Gauss map, discriminant locus, reduced system, conjugative radii of convergence.
Citation: Q.K. Phan, Series of Hypergeometric Type and Discriminants, J. Sib. Fed. Univ. Math. Phys., 2023, 16(4), 540-548. EDN: XIDJZF.
1. Introduction and preliminaries
Hypergeometric functions were studied in the 19th century by many famous mathematicians such as L.Euler, C.F. Gauss, E.Kummer, B.Riemann. Most of the researches were on one variable series. At the end of 19th and the first half of 20th century the hypergeometric functions were widespread considered, including several variables cases. Among them are the functions studied by G.Lauricella [11], J.Horn [8], P.Appell [3] (see also the books [4,5]). The hypergeometric functions are still attractive recently (see [2,6,13,14]). According to Horn [8] the series
H (X1,...,XN ) = ^ Ca (1)
aeNN
is called hypergeometric if the relations of neighboring coefficients
hi(a) = ^, i =1,...,^, (2)
ca
(where the set of ei composes the standard basis in ZN), are rational functions in variables a = (a1,... ,aN). Limit values of functions hi along fixed directions s = (s1,..., sN) G RN \ {0}
Pi(s) := lim hi(ks)
play an important role. We call the vector limit
1 ( 1 1 )
P(s) VPiW" ' Pn (s)J
the Horn parameterization or Horn uniformization for the hpereometric series (1). These vectors define the conjugative radii of convergence for the series (1) (about the conception of these radii see [16, Sec. 7, ch. 1]).
* phquangkhanh@gmail.com © Siberian Federal University. All rights reserved
In this paper we study the hypergeometric type series. Roughly speaking, these series satisfy the following conditions: there is a sublattice L c ZN of rank N such that the restriction of H on the shifts of L are hypergeometric. The details about the hypergeometric type series refer to the Section 3.
We are interested in the hypergeometric type series in order to investigate the solutions to universal systems of polynomial equations. In particular, we intend to apply the discriminant apparatus considered here to the calculation of the convergence domain of these series. Consider a general system of n polynomial equations with n unknowns y1,... ,yn:
Pi := a(V = 0' i = !,...,n, (3)
xeA(i)
where A(i) are the finite subsets of Zn and yx = y^1 .. .y„n. We assume that all coefficients a(i) are independent, and call (3) an universal algebraic system. Applying the Stepanenko's formula (see [10]) we get the hypergeometric type series presenting the monomials with positive integer exponents of the principal solution to the system (4).
We will explicit the relation between the Horn parameterization for these series and
p(s)
the parameterization ^ of the discriminant locus V of the system (4) (see more about ^ and V in Section 2.). According to result in [1], the parameterization ^ is the inverse of the logarithmic Gauss map for V. (The logarithmic Gauss map y : V c CN ^ CPN-1 for a hypersurface V, defined by polynomial P, can be defined by the formula
(zi, ...,zn) -—> (zidiP (z) : • • • : ZndnP (z)),
where dj is the derivative d/dzi (see [9,12])).
According to Kapranov's result in [9], the Horn parameterization — for the hypergeometric
series coincides with the parameterization ^ = y-1 of the discriminant locus V. The following theorem gives an extension of the Kapranov's result in [9] for the series of hypergeometric type representing monomials of solutions to the reduced system (4).
Theorem 1. The Horn parameterization for the series (6) and the parameterization ^(s)
p(s)
of discriminant set for the system (4) coincide:
* = !.
P
2. Reduced systems and their discriminants
Following the paper [1] we consider the reduced system of the system (3) in the forms
ym + E 4j)yX - 1 = 0, j = l,...,n, (4)
AeA(j)
where each mj is a positive integer and A(j) does not contain A = 0 and A = (0,..., mj,..., 0).
Denote by V0 the set of all the coefficients for which the system (3) has multiple zeros in the torus Tn = (C \ {0})n, i.e. the Jacobian of P equals zero. The discriminant locus V of the system (3) is the closure of the set V0 in the space of coefficients of polynomials P1,... ,Pn. Denote the matrix
A := (A(1),...,A(n)) = (ai,...,an),
where Ak = (Af,... ,Afn)T € A(j) are column-vector of exponents in monomials of equations (4).
Also let wm denote the n x n-diagonal matrix with values — on the diagonal. Consider the
mj
matrices
$ := wA, $ := $ - x,
where x is the matrix, whose i-th row is assigned by the characteristic function of the subset A(i) c A, i.e. elements of this row are 1 at the position A € A(i) and 0 at all positions A € A\ A(i). In addition, <pk denotes the rows of $, and pf denotes the rows of $. Their elements are denoted by fkx and pk\ correspondingly. We can interpret each row as a sequence of vectors
(1) (n)
We will follow two copies of CN. The first one is with the coordinators x = (x\), and the second one is with the coordinators s = (s\) constructed as a space with homogeneous coordinators for CPN_1. Following the result of Antipova and Tsikh (see [1]), the map
*: CPN-1 ^ CN = x-x C$:1,
from a projective space to the space of coefficients x = (x\) of the system (4), defined by
(j) n / , \
xj = "T^T n M 'A e A(j),j = 1'---'n' (5)
m,s) k=i\ <Pk, s) J
gives the parameterization for the discriminant locus V.
3. Solutions to reduced systems of algebraic equations
For the solution y = (y1,...,yn) to (4), we consider the series representing the monomial function y* = y11 ... y*:
y* = £ CaXa. (6)
aeNN
We focus on the so-called principal solution to system (4): they satisfy initial condition y(0,..., 0) = (1,..., 1). When Hj > 0 the Stepanenko's result [10] claims that the coefficients ca in (6) admit the following expression:
= (_i)«i+-+«N . ra • Ra, (7)
Ca
where
nr( ^ + < p j ,a))
j=i 3
N n '
nr^3+m+< pj>*)- s «o (8)
i=i j=i ieAU) v '
R„ = ,*(j - JP+j)
\i Hj + <Pj ,a) )
Vj + <Pj,a), ,..)cp P
(i ,j)ePaxPa
with Pa C {1,... ,n}. We call ra the gamma-part and Ra the rational-part of the coefficient ca. Remark that according to expressions (7) and (8) ca admits the expression
ca
M
ca = ta R(a) J} r(<aj, a) + bj), j=i
where ta = t^1 .. .tNN, ti,bi G C, aj G QN, and R(a) is a rational function. In the case when aj G ZN this expression presents the general coefficient Ore-Sato for hypergeometric series
(see [7,15]).
l'a
4. Horn parameterization for hypergeometric type series
Here we give more details for the definition of the series of hypergeometric type and construct for them the analog of the Horn parameterization. Let e1,... ,eN denote the standard basis of ZN, i.e. e\ = (0,..., 1,..., 0) with 1 being on A-th position. For a given v = (v1,..., vN) G (N\{0})N we consider the sublattice Lv c ZN generated by v1e1,..., vNeN. For two vectors v,s G ZN we define their product vs := (v1s1,..., vNsN).
Definition 1. We say that the power series
c«xa (9)
is of hypergeometric type if there exists v G (N \ {0})N such that all subseries
Hi := Y^ caxa = tV cStS, 1 G J,
ael+LvnNN s£NN
are hypergeometric in variables t\ = x^x, where c's = cl+vs and J is the sequence of all representatives for the factor ZN/Lv:
J = {(l1, ...,lN) G Zn : 0 < li < vi - 1,i = 1,...,N}.
The subseries Hl is hypergeometric iff all the relations
c
R(s) := , A = 1,..., N, (10)
c's
are rational functions of variables s = (s1,..., sN).
Proposition 1. The series (6) with the coefficient (7) is a hypergeometric type series. Proof. For
a vector v G (N \ {0})N we take v — (t, ..., t) where t is the least common multiple of m-1, .. ., mN.
According to (8), the relations (10) become
r (s)= ci+vis+ex) = ri+rs+rex (-1)trl+ts+tea , A = 1, . . . , N, l G J, (11)
cl+vs rl+Ts Rl+Ts
where J = {l = (l1,..., lN) : 0 < l1,... ,lN < t — 1}. The power of the exponent ( — 1)T comes from
(_1)|l + T (s + ea ) |
( 1)-= (-D1^1 = (_1)T
( —1)|l+vs| ( 1) ( 1) ,
where |a| := a1 + • • • + aN.
Here l + ts denotes the restriction of a on the shifted lattice l + Lv (i.e. a =: l + ts for some l G J). Thus rl+Ts and Rl+Ts are correspondingly the restrictions of the gamma-part ra and the rational-part Ra of the series (6) on the such lattice. It is clear that the second ratio in (11), the ratio for Ra, is a rational function in s. Introduce denotations
Ak := <Pk = (^k1,..., VkN), An+k := <fk = (<fk1,..., <fkN), A2n+A := ex, and rewrite (8) in such a way
J! r({Ap,a) + Vp) p=i
2n 2n+N
n r((Ap,a) + np ) n r((Ap,a) + 1)
p=n+1 p=2n+1
where np are some constants independing on a. To compute the ratio of gamma-parts in (11) we use the Pochhammer symbol
(z)k = = z(z + 1)... (z + k - 1), k G N\{0}
I(z)
„A .
and the denotation q2 := (Ap,re\). Then it leads to
n (iAp,a) + np - 1 +
r a+rex p=l
2n 2n+N
n ({Ap,a) + np - 1 + q^)qx n ({Ap,a) + q*)q
p=n+l p=2n+l
With a = l + ts, we get the ratio of gamma-parts restricted on the shifted lattice l + Lv
n
r n(
r l+rs+rex _ P=1
H((TAp,s) + n'p + q*U
2n 2n+N
n HTAp,s) + n'p + q*)qx n (t (ex,s) + /a + q£)qx
p=n+1 p=2n+1
(12)
where constants q'p are independent on s.
Since mj divide t, the delation tAp in (12) is a vector with integer coordinators. Then its
r
turns out that the relation l+TS+Tex in (11) is a rational function of the variables si, ... ,sn.
1 l + TS
Thus the series (6) is of hypergeometric type. □
According to Horn (see [8]) the convergence radii of hypergeometric series are defined by the limits
lim hi(rs), i = 1,..., N,
where the rational functions hi are defined by (2). In the hypergeometric type case, the convergence radii of the series (9) are defined by the limits
Pa(si,...,sn ) = lim (RA(rs))1, A = 1,..., N, (13)
where are rational relations (10) and t is the least common multiple of vi,...,vN, (si : • • • : sn) e RPn_1, si > 0. Indeed (si, ..., sn) are homogeneous coordinates in CPN_1, and the limits pi are rational and homogeneous of degree zero. They depend only on the ratio s = s1 : ••• : sn. The vector limit
1/1 1
:=
P(s) • 'Pn(s)J
are called by Horn parameterization (or Horn uniformation) for hypergeometric type series since Horn is the first person who considered such a limit for hypergeometric function (see [9]).
l'a
5. The proof of the Theorem 1
According to (12) we get the following formula for the limit values of relation (11) along direction s := (su..., s N ) G RN \ {0}.
Proposition 2.
PX(s1 ,...,3N ) = -< P ,S
<pj,s) TT ( PS) <ex,s) ¡¿\ < PpP, s) )
VpX
Proof. From the ratio (12) and the limits (13), Px(si, ...,sn )
where
lim Cl+Trs+Tex = lim
Cl+Trs
r l+Trs+Tex (-1)T Rl+rrs+rex
r
l+Trs
R
l + Trs
=: - lim [A-B-C
(a ■ B ■ cj
A :=
n (<r'Ap, s) + n'p + T )qx ¡=1
2n qx 2n+N qx
n (<rAp,s) + ri< + qT )qx n (r<ex,s) + T + T )q
¡=n+1
¡=2n+1
B
C
T qx+---+qx
T qn+i
q2n . tHn +
qn +
1+ +q2n+N
det (§(j) _ (v<i3),l(3)+TeX3)} + (v<i3) ,Trs(i)) 1 i Pi + (Vi ,l+Tex) + {<Pj ,Trs)
(i,j
(i,j)ePaxPa
(SÏ
det [j - ^
° H + ,l) + (Vj ,Trs) J(i,j)epaXpa
(i)
Recall that
Ak = ( Pk1, . . . , PkN ), An+k = (Ppk1,..., PpkN ), A2n+A = ex, qX = <Ap,rex), p G {1,..., 2n + N}.
Hp — v '-p
Since rex = (0,...,t,..., 0) with A €{1,...,N}
t Ppx
with 1 ^ p ^ n,
x = J tPf(p-n)x with n +1 < p < 2n, qp
T 0
with p = 2n + A,
with p > 2n and p = 2n + A.
Thus
qx + • • • + qn = (vix + • • • + Vnx^ ?2n+i + • • • + qL+n = ^
and with the notice that A € A(j) for some j,
qx+i +-----+ qXn = ({pix +-----+ pnx)T = (fix +-----+ vm _ 1)t.
The sums in (17) and (18) lead to
T (Vla +-----VV:\)T
B =
T (Vla +-----+V:>.-1)T ■ TT
Let r tend to the infinity we obtain the limits:
n A
U(Ap,s)qP p=i
1,
lim A =
r 2n
n <Ap,s)qp <ex,s)T
¡= n+1
(14)
(15)
(16)
(17)
(18)
x
P
lim C -
det SY" -
(j) (vjj) ,Ts(j))
(<Pj ,Ts)
(i,j)ePaxPa
det S(j) -
(j) _ (vij),rs(j))
1.
(i,j)€PaxPa
Thus
lim (A ■ B ■ cY
V /
E[(Ap,s)qp p=1
2n
(ел, s)T ft (Ap, s)qp
p=n+1
Substitute coordinators of the vectors Ap formulated in (15) and the value of qa in (16), then
the limit lim (A • B ■ C
r—^oo
(a ■ B ■ CjT equals
Пф,^
p=1
2n
(ел,8)Т П (Фр-п,$)Тф(—)
p=n+1
1
In the square brackets each factor is an exponentiation with the power t . The radical — applying
t
on the square brackets leads to a simpler expression for the limit:
П ( фр, s)^ p=1
2n
(ел,з) П ( <fp-n,s)ï(p-n»
p=n+1
Rewrite the index for the production in the denominator of the last expression, it will become
(Vj ,s)U(<P P,s)VpX _p=_
n .
p=1
Combining the factors with the same index under the production signs in the numerator and in the denominator of the last expression we will get the result:
(<fj,s) тт (
(ел, s) Щ (hp,s)J
ФрЛ
p=1 w ^
Consequently we get the formula for the limit Pa:
(h,s) n ( фп^л
(ел, s) УД (hp,s)J
фрЛ
The proposition holds.
Now we are ready to prove the Theorem 1.
□
л
Proof of theorem 1. From (5) and (14) it turns out that
4j) = , x G aU)>i
Pa
Thus the parameterization ^(s) for the discriminant locus V of the system (4) composed by the coordinators x\j) coincides with the limit vector of the hypergeometeric type series (6) composed
by the coordinators ——. The theorem holds. □
Px
I would like to express my deepest gratitude to professor Avgust K. Tsikh. He directed me and helped me a lot with completing the paper. I am also grateful for the financial support of the Krasnoyarsk Mathematical Center and the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2023-936).
1
References
[1] I.A.Antipova, A.K.Tsikh, The discriminant locus of a system of n Laurent polynomials in n variables, Izv. Math., 76(2012), no. 5, 881-906. DOI: 10.1070/IM2012v076n05ABEH002608
[2] K.Aomoto et al., Theory of Hypergeometric Functions, Springer Monographs in Mathematics. Springer Japan, 2011.
[3] P.Appell, On hypergeometric functions of two variables, Resal J, 8(1882), no. 3, 173-216.
[4] W.N.Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 32, 1964.
[5] H.Bateman, A.Erdelyi, Higher transcendental functions 1, New York: McGraw-Hill, 1953.
[6] A.N.Cherepanskiy, A.K.Tsikh, Convergence of two-dimensional hypergeometric series for algebraic functions, Integral Transforms and Special Functions, 31(2020), no. 10, 838-855. DOI: 10.1080/10652469.2020.1756794
[7] I.M Gelfand, M.I.Graev, V.S.Retakh, General hypergeometric systems of equations and series of hypergeometric type, Uspekhi Mat. Nauk, 47(1992), no. 4, 3-82 in Russian.
[8] J.Horn, Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veraderlichen, Mathematische Annalen, 34(1889), 544-600.
[9] M.M.Kapranov, A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann., 290(1991), 277-285.
[10] V.R.Kulikov, V.A.Stepanenko, On solutions and Waring's formulas for systems of n algebraic equations for n unknowns, St. Petersburg Mathematical Journal, 26(2015), no. 5, 839-848. DOI: 10.1090/spmj/1361
[11] G.Lauricella, Sulle funzioni ipergeometriche a piu variabili. Italian, In: Rend. Circ. Mat. Palermo, 7(1893). 111-158. DOI: 10.1007/BF03012437.
[12] M.Passare, A.K.Tsikh, Amoebas: their spines and their contours, Contemporary Mathematics, 377(2005), Eds. G. L. Litvinov, V.P. Maslov ; AMS. Vienna, Austria: Idempotent Mathematics and Mathematical Physics: Intern.
[13] T.M.Sadykov, A.K.Tsikh, Hypergeometric and algebraic functions of several variables, Nauka, Moscow, 2014 (in Russian).
[14] G.A.Sarkissian, V.P.Spiridonov, Rational hypergeometric identities, Funct. Anal. Appl., 55(2021), no. 3, 250-255. DOI: 10.1134/S0016266321030096
[15] M.Sato, T.Shintani, Theory of prehomogeneous vector spaces (algebraic part) - the English translation of Sato's lecture from Shintani's note, Nagoya Mathematical Journal, 120(1990), 1-34. D0I:10.1017/S0027763000003214
[16] B.V.Shabat, Introduction to Complex Analysis. Translations of mathematical monographs. American Math. Society, 1992. DOI: 10.1090/mmono/110.
Ряды гипергеометрического типа и дискриминанты
Куанг Хань Фан
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Одночлен решений редуцированной системы алгебраических уравнений представляет собой ряд гипергеометрического типа. Мы распространяем результат Хорна-Карпранова (для гипергеометрических рядов) на случай рядов гипергеометрического типа.
Ключевые слова: ряды гипергеометрического типа, логарифмическое отображение Гаусса, дис-криминантное множество, редуцированная система, сопряженные радиусы сходимости.
