CC BY
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УДК 517.55
Discriminant and Singularities of Logarithmic Gauss Map, Examples and Application
Bernd Martin*
Institute of Mathematics, Brandenburg University of Technology Cottbus, PF 101344, 03013 Cottbus Germany
Dmitry Yu. Pochekutov^
Institute of Core Undergraduate Programmes, Siberian Federal University, Svobodny, 79, Krasoyarsk, 660041,
Russia
Received 24.08.2012, received in revised form 06.10.2012, accepted 06.11.2012 The study of hypersurfaces in a torus leads to the beautiful zoo of amoebas and their contours, whose possible configurations are seen from combinatorical data. There is a deep connection to the logarithmic Gauss map and its critical points. The theory has a lot of applications in many directions.
In this report we recall basic notions and results from the theory of amoebas, show some connection to algebraic singularity theory and consider some consequences from the well known classification of singularities to this subject. Moreover, we have tried to compute some examples using the computer algebra system Singular and discuss different possibilities and their effectivity to compute the critical points. Here we meet an essential obstacle: Relevant examples need real or even rational solutions, which are found only by chance. We have tried to unify different views to that subject.
Keywords: logarithmic Gauss map, singularities, discriminant, asymptotics, hypersurface amoeba.
1. Toric Hypersurface and Logarithmic Gauss Map
Let V*(f) be an algebraic hypersurface in the algebraic torus Tn, T := C*, i.e.
V*(f) = {z e Tn | f(z) = 0},
where f (z) = ^ aaza is the Laurent polynomial.
AcZn
Recall that the Newton polyhedron Nf C Rn of f is the convex hull in Rn of Af := supp(f). Let Xs be the smooth toric variety associated to the fan E, which is a refinement of the fan dual to the Newton polyhedron Nf. We denote by V(f) C Xs the closure of V*(f) in Xs. The polynomial f is called non-singular for its Newton polyhedron if V*(f) is smooth and for any face A C Nf one has
(zidf (A)/dzi,...,z„df (AVdz„)=0
for all z e V(f) nXS', where f(A) is the truncation of f to the face A, and is the toric variety associated to A. In accordance with singularity resolution theorem, cf. [6, page 291], a generic polynomial f is non-singular for its Newton polyhedron, and, therefore, V(f) is non-singular.
* martinb@tu-cottbus.de tpotchekutov@gmail.com © Siberian Federal University. All rights reserved
Next we introduce the so-called logarithmic Gauss map Yf : V*(/) ^ Pn-1. Let tn denote the Lie algebra of Tn which is identified with the tangent space of Tn at the unit point e. For any point z e V* shift the tangent space Tz (V*) by the torus multiplication (with z-1) to a hyperplane hz c tn, inducing a point in the projective space of the dual tn*, which we define to be Yf (z) := h* e Pn-1 := P(tn*). In coordinates of Tn the map Yf is given by
z
mn—1
Yf(z) = (z/ : ... : z/) G P"
Described in more geometric terms we have: Let U C Tn be a neighbourhood of a regular point z on V*(/). Choose a branch of the logarithmic map (restricted to U) log : U ^ Cn then the direction of the normal line at log(z) to transformed hypersurface log(V*(/) n U) has components (z1/zi,..., zn/Zn). This construction does not depend on the choice of the branch of log.
In [11, Section 3.2] one can find the idea of a construction how to extend Yf in the non-singular case to a finite map
Yf : V(/) ^ Pn-1.
Having a finite map y to a smooth variety, one can associate the ramification locus or the discriminant as image of the critical locus: D := y(Cy), which is usually a hypersurface. An analytic structure which is compatible with base change was introduced by Teissier, cf. [14]: The structure sheaf OD is defined to be the quotient by the 0-th fitting ideal of y*(Oc). In local coordinates the defining equation is obtained as the (classical) discriminat of the polynomial, that generates the finite extension of the structure sheafs over an open affine subsets. From the well-known theorem of Kouchnirenko, cf. [7, Th. 3], Mikhalkin obtains:
Proposition 1 ( [11]). If the polynomial / is non-singular for its Newton polyhedron, then the degree of Yf is obtained as
deg(Y7) = n! • Vol(Nf).
For later calculation we give a description of the logarithmic Gauss map Yf in local coordinates. Since V*(/) is smooth, we assume w.l.o.g. that locally /Zn := d//dzn = 0. Then there exists a function g(z'), z' := (z1,... ,zn-1) such that /(z',g(z')) = 0
Since gzi = /(z',g)//zn(z',g) and (log(g(z'))z¿ = gz¿/g hold, one obtains the formula
, ( d log g(z') d log g(z') \
Then the fiber y- 1(y), y = (yi : • • • : G Pn 1 is given by the zeros of the local complete intersection ideal generate by f and the 2-minors of
Zi/zi • • • znfzn
yi •• • yn
1.e. (in case of yn = 0) y-1 (y) is defined by the complete intersection ideal
Iy := (f, hi, • • • hn-i), (2)
where hj = ynzjfzi — yjznfzn. There are at most n! • Vol(N/ ) zeros in the torus by Proposition 1.
2. Amoeba and its Contour versus Laurent Series
Consider a rational function F(z) = h(z)/f (z) of n complex variables and different Laurent expansions
E c«z° (3)
aez
of F centered at z = 0. The most natural way to describe these expansions uses the amoeba of polar hypersurface V* = V*(/).
Recall that the amoeba Av* of a toric hypersurface V* = V*(/) is the image of V* by the logarithmic map Log : Tn ^ Rn
Log :(z!,... 7 Zn ) ^ (log |zi log | zn 1 ).
The complement Rn — Av* to the amoeba consists of a finite number of connected components Ej, which are open and convex, cf. [4, Section 6.1]. These components are characterized in the following Theorem, which is a summary of Propositions 2.5, 2.6 in [3], Theorem 10 and Corollary 6 in [13, Section I.5].
Theorem 1. There exists an open subset Un in the set of polynomials with fixed Newton polyhedron N that satisfies the following property:
If / £ Un, then there is a bijection from the set of lattice points of Nn Zn to the set of connected components of Rn — Av* (f) : v ^ Ev such that the normal cone Cv(v) to Nf at the point v is the recession cone of the component Ev.
A recession cone is the maximal cone which can be put inside Ev by a translation. If / £ Un, the expected component Ev may not exist for some non-vertice lattice points v £ N because the associated Laurent series below does not converge.
Given a component Ev one obtains a Laurent series of F centered at z = 0 using the term av zv of / as denominator in a corresponding geometric progression
1 - (avzv — /)k
/ ^ (avzv)k+1 ' J k=0 v v '
(4)
The set {Log-1(Ev)} contains the domain of convergence for this Laurent series. The support of expansion (4) is the minimal cone Kv that after a translation by v contains the face A c Nf, which has v as an interior point.
A non-zero vector q £ Zn n Kv defines a so-called diagonal subsequence {ck.q}keN of the set of coefficients of expansion (3). We will discuss its asymptotics in the next section.
The set of critical values of the map Log restricted to V* is called the contour Cv* of the amoeba Av* (see [12]). The contour is closely related to the logarithmic Gauss map Yf. Recall Lemma 3 from [11].
Lemma 1. The preimage of the real points under the logarithmic Gauss map is mapped by Log to the contour:
Cv * = Log (y-1«-1 )) .
Proof. Let z be a regular point on V* and U its neighbourhood. Since the map Log|v* is a composition of log : z ^ (log(z1),..., log(zn)) and the projection Re : Cn ^ Rn, the point z is critical for Log|v* if the projection d Re : Tz log (V* n U) ^ Rn is not surjective at z. A fiber Tz log (V* n U) of the tangent bundle of the image by log of the hypersurface V* is the hyperplane
{t £ Cn : (Yf (z),t> =0}.
For real Yf (z) the projection d Re is not surjective. If Yf (z) is not real one can consider (Yf (z), t> = 0 as a system of linear equations with fixed real part Re t, and solve it with respect to Imt. Hence, z is not critical for Log|v* . □
Therefore, the contour Cv* can be computed as the Log-image of the zeros of the ideal
(/, qnz/ — q1zn/z„,... , qn zn— 1 /*„_! — qn—1 zn/z„ ^ (5)
where q runs through all real points (q1 : ... : qn) £ PR-1, (here w.l.o.g qn = 0).
3. Singularities of Phase Function
Consider the function
$ : Pn-1 x V* —► C, $(y, z) = (y, log z) .
Introduce the phase function yq := $(q, —), later we show that it is indeed a phase function of some oscilllating integral. Denote by Crit(yq) C V* the set of critical (or stationary) points of function yq. It coincides with the preimage of the logarithmic Gauss map y/:
Proposition 2. The relative critical locus of $ coincides with the graph of Y/ :
CritPn-i ($) = r7/.
Proof. Assume fZn = 0 then we use local coordinates z' on V* and consider the function g(z') such that f (z', g(z')) = 0. We obtain
d$(z, y)/dzi = — +—yn— dg(z')/dzi, i = 1,..., n — 1. zi g(z')
Up to a non-zero constant multiple the components of the gradient d$(z, y)/dz together with the defining polymonial f (z) of V* give us the defining ideal (5) of the fiber of y by the logarithmetic Gauss map 7. □
The last statement shows us that the Log-image of Crit(yq) is contained in the contour CV* of the amoeba AV*, and the tangent hyperplane to CV* at a point Log(z0), z0 G Crit yq, has normal vector q G Zn — {0}.
Another consequence of the above formula concerns the connection between the singularities in the fibers of the phase function and the fibers of the logarithmic Gauss map:
Proposition 3. Let (z0,y0) G rYf be a point of the graph of 7/. Then the Jacobian matrix of Y/ at z0 coincides with the Hesse matrix of yy0 at z0 up to multiplication with a regular constant diagonal matrix D:
Hess(y>y0 )(z0) = D • Jac(Y/)(z0).
Proof. As before we assume fZn (z0) = 0 and use local coordinates z'. From (1) we obtain the entries of the Jacobian matrix Jac(Y/) of the map y/
T / n ( d2 log(g(z')) d log(g(z'))\ .... n
Jac(Y/)(i'j) = — (,zi 3zi3zi + Si* dzj J'^ = 1'...'n — 1'
where 5ij is the Kronecker symbol. Moreover,
d2yy = d2 log(g(z')) yi_ dzidzj n dzidzj ij z2
holds for the second derivatives of yy. Since = —y0 n d log(g(z0)) at a critical point z0 of
z0,i ' dzi
yy0, we obtain the statement by putting the i-th entry of D to be di = — . □
zi
We obtain as corollary of the last proposition that for directions y = q outside the ramification locus of the logarithmic Gauss map y/ the phase function yq has only Morse critical points.
Corollary 1. The logaritimic Gauss map y/ is unramified at q G Zn — {0} iff the phase function yq has only Morse critical points, e.g. non-degenerated singularities.
Proof. The map Yf is not ramified over y iff its Jacobian has full rank at all points of the fiber Y-1(y). From Proposition 2 we get
det iPrr(zo)) = -—-det(Jac(7f)(zo)).
\dzidzj J zo,i ••• zo,n-i
Hence, the Jacobian determinant does not vanish iff the Hessian is not zero at corresponding points. □
Next we want to discuss degenerated critical points of the phase function. By Mather-Yao type theorem the R-class (right-equivalence) of an analytic function h(z) e C{z} =: On at an isolated critical point z0 =0 is equivalent to the isomorphy type of the Milnor algebra Qh := On/(dh/dz), but as C[t]-algebra, the action of t on Qh induced by multiplication with h, cf. [10]. The isomorphy type of the associated singularity (V(h),z0), i.e. the K-class (contact-equivalence) of h(z), is equivalent to the isomorphy class of the Tjurina algebra Th = Qh/(h) itself, cf. [9]. Obviously, these equivalence classes coincide for quasi-homogeneous functions (because ^(h) = t(h), Th = Qh, hQ = 0). The Milnor algebra of the phase function at z0 coincides with the local algebra of the fiber Y-1(Yf (z0)) at z0.
Corollary 2. If (z0,y0) e rYf, denote by Qv the Milnor algebra of the function (pyo (z) — y0) at z0, then we have
Qv = O7/-1(yo),zo and QV/Ann(mQ) = OSi„s(77l(y0)),z0 •
Proof. By Proposition 2 the germs coincide: (Crit(pyo ),z0) = (y-1 (y0),z0). The algebra of the critical locus is the Milnor algebra of (pyo (z) — y0). By Proposition 3 the Jacobi determinant of Yf at z0 equals up to a constant multiple to the Hessian of pyo at z0, which generates the annulator of the maximal ideal in the local complete intersection algebra Qv. □
A function h e On with isolated critical point is called almost quasihomogeneous, if ^ = t + 1. This is equivalent to hQh = Ann(mq). Assume that the singularities in a fiber of a phase function are quasihomogeneous or almost quasihomogeneous, then in spite of Mather-Yao type theorems these singularities are determined by the fiber germs of the logarithmic Gauss map because or Qv/(Ann(m)) = Tv, respectively. Note, that all simple or unimodal critical points belong to these singularities. The singularities of a phase function on their part determine the asymptotic of corresponding oscillating intergrals.
All degenerated critical points are lying over the singularities of the discriminant D C Pn-1 of Yf. Many results could be found concerning the connection between singularites of discrimant and singularities in the fiber. We try to discuss some consequnces with respect to our setting.
The finite map Yf can be considered as family over Pn-1 of complete intersections (of relative dimension zero). Let (X0,0) be a germ of an isolated complete intersection singularity, let X ^ S its versal family with discriminant D C S, the singularity of the discriminat (D, 0) determines the special fiber (X0,0) up to isomorphy by a result of Wirthmuller, c.f [17]. If dim(X0) = 0 the multiplicity of the discriminant fulfills mult(D, 0) = dimC(OXo) — 1 = dimc(OSing(Xo)), as a consequnce of [8], for instance. This is globalized straight forward.
Proposition 4. Let y : X ^ S be a finite morphism with discriminant D C S and each Xs is a complete intersection, then holds:
mults(D) mult(Sing(Xs), zi) = ^ (mult(Xs, zi) — 1).
Zi£Xs Zi£Xs
Moreover, equality holds at s e S, if y induces a versal deformation of Xs.
Proof. The local branches of D at s are corresponding to the discriminants Di of the germs (X, zi) ^ (D, s), hence the multiplicities of Di add up to the multiplicity of D. Any family is locally induced from a versal one, hence the discriminant is induced by base chance from the discriminant of the versal family and its multiplicity cannot become smaller. □
Note, versality is an open property and corresponds to some kind of stability in the sence of Mather. It is not clear for us, whether (or under which additional assumtions) the logarithmic Gauss map Yf for a generic function f (z) with fixed Newton polyhedron N has this stability property. It holds in all computed examples. But, an answer needs further investigation.
Inspecting the classification of hypersurface singularities we get the types of possible critical points for small multiplicities of the discriminant, which are listed in the following Corollary.
Corollary 3. Given a Laurent polynomial f (z), non-singular with respect to its Newton polyhedron, and let y be the corresponding logarithmic Gauss map with discriminant D C Pn-1. Let m = m(q) := mult(D, q), then the following configurations are met for the fiber Fq := y- 1(q), respectively for the collection of critical points of the phase function pq(z):
• m =1: Fq has exactly one point z* of multiplicity 2, pq has non-degenerated critical point and one A2-singularity at z*.
• m = 2: Fq either one point of multiplicity 3 or at most two points of multiplicity 2, pq has at most one A3 or two A2-points.
• m = 3: Besides A1 can occur the following collections of critical points of pq: one D4 or one A4 or a combination k2A2 + k3A3 with k2 + 2k3 ^ 3.
• m ^ 6: Type of critical set of pq: Only (simple) ADE-critical points can occur
8
kiAi + J2 hDi + J2 niEi,
1 i=6
such that
y^ i(ki + li + ni) < n! vol(N)
and
y^(i - 1)(ki + li + ni) < m.
• m ^ 6: all critical points are quasihomogeneous (and simple or unimodal).
• m ^ 14: all critical points are almost quasihomogeneous (and simple or unimodal).
The first critical point, which is not almost quasihomogeneous are the bimodal exceptional singularities with smallest Milnor number ^ =16 of type Q16 or U16, cf. [1]. They can occur only at multiplicity m ^ 15.
4. Representation of Diagonal Coefficient by Oscillating Integrals and its Phase Function
In this section we return to Laurent series (3) converging in Log-1(Ev). We explane the residue asymptotics formula for its diagonal coefficient in the direction q e Zn n Kv.
Recall that the Laurent series coefficient can be represented in the form
= 1 f "
(2ni)n za+1'
where w := F(z)dz and the cycle rv is n-dimensional real torus Log (xv), xv G Ev. The direction q induces series of diagonal coefficients
cVk = (2!)^ JTv z^w+r • (6)
We may assume that the point xv generates a line L := Rxv c Rn which is transversal to the boundary and intersects it at a point p, and the normal vector at p to coincides with the vector q. In other words, p is the Log-image of points w(1)(q),..., )(q) from the fiber 7-1(q) of the logarithmic Gauss mapping. The torus Log-1(p) c Log-1(L) intersects the hypersurface V* at most in N < n! • Vol(Nf) points.
Consider a heighbourhood Ui in Cn of the point w(i)(q), then Log-1(L) intersects the hypersurface V* in Ui along an (n — l)-dimensional chain hi c V*. It can be shown, cf. [15] for the case n = 2, that integral (6) is asymptotically equivalent for k ^ to the sum
<-k = I res (W) • e-^^z>'k, (7)
where log z = (log z1,..., log zn) and res (w/z) is the residue form. In local coordinates z' = (z1,..., zn-1) of V* (assuming fZn = 0) we have res (W) = f —. Therefore, the
z z JZn \v*
diagonal coefficient can be represented as the sum of oscillating integrals with the phase function yq(z') = (q, log z)\V*. The critical points of this phase function give the main contribution to the asymptotic of such integrals. From Proposition 2 follows that the support of hi contains only one critical point of yq. It is a point w(i)(q) G Y-1(q).
The asymptotics of an oscillating integral is the most simple for Morse critical points. In this case it is given by stationary phase method (also called saddle-point method, see [18]). The Corollary 1 of Proposition 3 states that for directions y = q outside the ramification locus of the logarithmic Gauss map 7 the phase function yq has only Morse critical points.
The situation of a degenerated critical point is much more complicated. First of all we are looking only for rational critical points! By a result of Varchenko some information about asymptotics of oscillating integral can be read from the distance of the Newton diagramm of the phase function at the corresponding point in case of a Newton non-degenerated phase function (and then it depends only of the K-equivanence class of the hypersurface singularity). Otherwise, the distance is only a lower bound. So called adapted coordinates exist always in dimension 2 such that the phase function is Newton non-degenerated. Adapted coordinates can be computed algorithmically, for more details cf. [5] and [16].
5. Discussion of examples
Example 1. Consider the smooth hypersurface V*(/) defined as a zero set of the polynomial
f = z2z2 + - Z1Z2 + a, a G R, a = 0, ,
which is non-degenerated for its Newton polyhedron. The cubic V*(/) is a two-dimensional real torus with three removed points.
The solutions z(y) = (z1(y),z2(y)) of
zfz2 + z1 z>2 — z1z2 + a = 0, h := (2y2 — y1)z2z2 + (y2 — 2y1)z1z| + (y1 — y2>1 z2 = 0
for fixed parameter (y1 : y2) e P1 are zeroes of ideal (5), and for real parameter y, they are projected to the contour CV* by Log-map. We are interested in the real ramification locus of Yf. We compute the resultant of f, h with respect to the variable z2
Res(f, h) := (-y2 + ym + 2y2)z? + (2y2 - 2y^ - y22)z2 + (-y2 + y1y2)z1 + 4ay2 - 4ay1y2 + ay2.
The multiplicity of an isolated zero z(y) of system (8) coincides with the multiplicity of the zero z1(y) in Res(f, h). The discriminant of the polynomial Res(f, h) with respect to z1 is the homogeneous polynomial
D(y1,y2) = (1 - 27a)(-2y1 + y2)2(4ay6 - 12ayfy2 + (-3a + 1)y4y|--2(1 - 13a)y?y23 + (-3a + 1)y2y4 - 12ay1y25 + 4ay26)
in variables y1, y2.
Interested in roots of (8) in T2 we can omit the factor (-2y1 + y2)2 in the last expression. Substituting in D(y) an affine parameter A = y1/y2, we get the polynomial
D(A) = 4aA6 - 12aA5 + (-3a + 1)A4 - 2(1 - 13a)A2 + (-3a + 1)A2 - 12aA + 4a,
whose real zeroes Aj give the points (Aj : 1) G PR of the real ramification locus of Yf •
We have three real intervals of the parameter line Ra: for a < 0 the polynomial D(A) has six
real roots, for 0 < a < — and — < a the polynomial D(A) has no real roots.
Choosing values of a from the different intervals of Ra, we obtain different configurations of the contour CV* and the amoeba AV* • Because the volume 2! • Vol(Nf ) = 3 does not depend on a, all these configurations have a following common property: The number of preimages Log-1(p) of a point p G CV* with normal vector (y1 ,y2) G R2 is equal to three. We count such preimages, which are solutions to (8) for corresponding (y1 : y2) G PR, with their multiplicity in (8). Hence, one can find for every A G R three points on CV* with the normal vector (A, 1). Moreover, each one lies on its own colored or black part of the contour (see Fig. 1).
Fig. 1. The contour and the amoeba (shaded) for the polynomial f = z2z2 + z1z| - z1z2 + a: on the left a < 0, in the middle 0 < a < 1/27,on the right a > 1/27
On the left Fig. 1 six black points on CV* are images of pleat singularities of the mapping Log|V* ; they correspond to values Aj that belong to the real ramification locus of Yf.
Although for a > 0 the real ramification locus of Yf is empty, we can distinguish two situations. If 0 < a < 1/27 the hypersurface V*(/) is a complexification of the so-called Harnack curve and the complement of its amoeba has the maximal number of components. In this case Log|V* has
only fold singularities, which coincide with V* (/) n R2. For a > 1/27 the complement of the amoeba Av* has no bounded, component and the mapping Log|v* has three pleat singularities; other singularities are folds.
Therefore, for a > 0 Yf-fiber of any rational A contains only the Morse critical points of the phase function. For example, set the parameter a = 3/100 then the Yf -fiber of A = 1/3 consists of the Morse points (3/10,1/2), (7/40 + N/57/40,9/8— n/57/8) and (7/40-N/57/40, 9/8 + n/57/8). For a < 0 we can get degenerated rational points in a real ramification locus, e.g. there are six rational points —2, —1/2,2/3,3/2,1/3,3 in the real ramification locus of Yf, a = —9/10. The Yf -fiber of such points has a simple point and an A2-point of the phase function.
Example 2. We consider the polynomial / in n = 3 variables, which is non-degenerated for its Newton polyhedron;
/ = 1 + z1 + z2 + z3 + 3z1 z2 + 3z1 z3 + 3z2 z3 + 11z1 z2z3.
As in Example 1 the real ramification locus of Yf is determined by the following system
1 + z1 + z2 + z3 + 3z1z2 + 3z1z3 + 3z2z3 + 11z1z2z3 = 0, y3z1 — y1 z3 + 3y3 z1z2 + (3y3 — 3y1)z1 z3 — 3y1z2z3
< +(11y3 — 11y1 )z1z2z3 = 0, (9)
y3 z2 — y2z3 + 3y3 z1 z2 — 3^1 z3 + ( — 3^2 + 3^3 )z2z3
+ (11y3 — 11y2 )z1z2z3 = 0.
With similar computations we obtain the discriminant D(y) of the logarithmic Gauss map:
D(y) := • (^2 — y3)2 • (4y1 + 5y2 + 5y3)2 • d(y), (10)
where d(y) is a homogeneous polynomial of degree 12, it consists of 91 terms. Its Newton's polyhedron is a triangle with vertices (12,0,0), (0,12,0) and (0,0,12).
We do not consider zeroes of the first three factors in (10) because they do not give us multiple roots of (9) in the torus. The ramification locus of Yf is given by zero set of d(y). Let A1 = y1/y3, A2 = y2/y3 be coordinates in affine part of PR, where y3 = 1. Fig. 2 depicts the zero set of d(A1, A2,1) = d(y)/y12, which coincides with the affine part of the real ramification locus of Yf •
Fig. 2. The real ramification locus of Yf
The red points (1/9,1/9), (1/3,1/3), (1,3), (3,1), (1,9), (9,1) on Fig. 2 are degenerated rational critical points of the discriminant with Milnor number / = 2. This example of the polynomial / is a special one because the existence of rational degenerated points in a real
ramification locus is not a generic property. We are interested in such points because they lead to degenerated critical points of a phase function. In this example the Yf-fiber of any A2-point contains excatly one A3-critical point of the phase function (see Appendix for details).
Appendix. Computation with Singular (Some Experiences)
The computer algebra system Singular, cf. [2], was used for the computation of examples. We tried several strategies for computing the discriminant of the Log-Gauss map with different success, i.e. to get a result for non-trivial examples without overflow and in reasonable time. Here we give a small introduction how proceed in Singular, demonstrated with the equation of Example 2.
Start with a base ring that contains the ideal I of the graph of the log Gauss map y/ of a polynomial f = f (z) and compute I, (here n = 3):
ring R=0,(y1,y2,y3,z1,z2,z3),dp;
poly f=1+z1+z2+z3+3*z1*z2 +3*z1*z3+3*z2*z3+11*z1*z2*z3;
matrix A[2][3] = z1*diff(f,z1),z2*diff(f,z2),z3*diff(f,z3),y1,y2,y3;
ideal I = f,minor(A,2);
Next we project the graph restricted to some affine chart U3 := {y3 = 0} into A3 := U3 x A1 (A1 — a coordinate axes of A3). The image is a hypersurface defined by the next polynomial h(yi,y2, z1), which we could closure in P^ by homogenizing in the y's. Using the elimination of variable, the multiplicities of multiple factors may be lost, but it does not effect the result.
I = subst(I,y3,1); ideal J = eliminate(I,z2*z3); poly h1 = J[1];
The choice of the projection direction was good, if degz1(h1) = n! • Vol(N) = 6. The discriminant variety of y/ is contained in the discriminant hypersurface of the projection V(h1) C U3 x A1 —► U3, computed in the next step.
poly d1 = resultant(h1,diff(h1,z1),z1);
d1 = homog(d1,y3);
list Ld = factorize(d1);
The plane curve V(d1) C P2 has several components, it may have components with certain multiplicities, some of them induced from the closure V* (f) or not belonging to the discriminant. If our polynomial is generic, then we expect the discriminant of Y/ (i.e. restricted to the torus) being irreducible. We should test which factor is correct. Some components of V(d1) have empty fiber with respect to Y/ or no multiple points in its Y/-fibers. We can reduce sometimes the number of factor as follows: Compute for any coordinate zi (as above for i = 1) polynomials hi and di and factorize only d := gcd(d1,..., dn).
Having found the equation of the discriminant polynomial d0(y), we can compute its (discrete) singular locus.
poly d0 = Ld[1][2]; (choose the right factor in this example)
d0 = subst(d0,y3,1);
ring S = 0,(y1,y2,y3),dp;
poly d0 = imap(R,d0);
ideal sl = slocus(d0);
list Lsl = primdecGTZ(sl);
Here, the singular locus has six rational double points Q1 = (1,3), Q2 = (9, 9), Q3 = (1,9), = ( 1,1 ), Qs = (9,1), Q6 = (3,1) and more irrational singular points. We choose Q1 and check, that it is an A2-singularity of D.
show(Lsl[2]); ring S' = 0,(y1,y2),ds; poly d0 = imap(R,d0); d0 = subst(d0,y1,y1+1); d0 = subst(d0,y2,y2+3); "mu =",milnor(d0);
(choose one of the singular points of D)
(translate that singularity to zero) (Milnor number of the singularity)
Compute the 7/-fiber of Qi. It has 3 simple points and exactly one point P* = (-1, — ^, —1) of multiplicity 3, being an A3-point of the phase function.
setring R;
I = subst(I,y1,1); I = subst(I,y2,3); ring R0 = 0,(z1,z2,z3),dp; ideal I = imap(R,I);
list Lfib = primdecGTZ(i); (list contains the points of the fiber).
option(redSB);
show(std(Lfib[1][2]));
"mult =",vdim(std(Lfib[1][1]));
Similar computations lead to similar results at the other 5 rational singularities of the discriminant.
Authors are supported by the research school 'Hybrid Systems' of BTU Cottbus.
References
[1] V.I.Arnold, S.M.Gusein-Zade, A.N.Varchenko, Singularities of differentiable maps, Vol. I, Boston, Bikhauser, 1985.
[2] W.Decker, G.-M.Greuel, G.Pfister, H.Schonemann, Singular 3-1-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2011).
[3] M.Forsberg, M.Passare, A.Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. in Math., 151(2000), 45-70.
[4] I.Gelfand, M.Kapranov, A.Zelevinsky, Discriminants, Resultants and Multidimentional Determinants, Boston, Bikhauser, 1994.
[5] I.A.Ikromov, D.Miiller, On adapted coordinate system, Trans. Amer. Math. Soc., 363(2011), 2821-2848.
[6] A.G.Khovanskii, Newton polyhedra and toroidal varieties, Funct. Anal. Appl., 11(1977), no. 4, 289-296.
[7] A.G.Kouchnirenko, Polyedres de Newton et nombres de Milnor, Invent. Math., 32(1976), 1-31.
[8] D.T.Le, Calculation of Milnor number of isolated singularity of complete intersection, Funk. Anal, and its Appl., 8(1972), no. 2, 45-49.
J.N.Mather, Classification of Isolated Hypersurface Singularities by Their Moduli Algebras, Invet. math., 69(1982), 243-251.
B.Martin, Singularities are determined by their cotangent complexe, Ann. Global Anal. Geom., 3(1985), no. 2, 197-217.
G.Mikhalkin, Real algebraic curves, moment map and amoebas, Ann. of Math., 151(2000), no. 1, 309-326.
M.Passare, A.K.Tsikh, Amoebas: their spines and their contours, Contemporary Math., 377(2005), 275-288.
H.Rullgard, Topics in geometry, analysis and inverse problems, Ph. D diss., Stockholm University, 2003.
B.Teissier, The hunting of invariants in the geometry of discriminants. Real and complexe singularities, Oslo 1076, 565-678.
A.K.Tsikh, Conditions for absolute convergence of the Taylor coefficient series of a mero-morphic fucntion of two variables, Math. USSR Sb, 74(1993), no. 2, 337-360.
A.N.Varchenko, Newton polyhedra and estimates of oscillating integrals, Funct. Anal. Appl., 10(1976), no. 3, 175-196.
K.Wirthmiiller, Singularities Determined by Their Discriminant, Math. Ann., 252(1980), 237-245.
R.Wong, Asymptotic approximations of integrals, SIAM, 2001.
Дискриминант и особенности логарифмического отображения Гаусса, примеры и приложение
Бернд Мартин Дмитрий Ю. Почекутов
Изучение гиперповерхностей, заданных в торе, приводит к прекрасному зоопарку амеб и их контуров, возможные конфигурации которых читаются из комбинаторных данных. Существует глубокая связь между теорией амеб и логарифмическим отображением Гаусса, а также его критическими точками, изучение которых находит приложения в различных областях.
В статье мы напоминаем основные понятия и результаты из теории амеб, раскрываем некоторые ее связи с алгебраической теорией сингулярностей. Более того, мы приводим вычисления критических точек логарифмического отображения Гаусса в системе компьютерной алгебры SINGULAR, а также обсуждаем различные варианты и их эффективность. Здесь мы приходим к существенному наблюдению: содержательные примеры требуют наличия вещественных или даже рациональных решений соответствующей системы алгебраических уравнений.
Ключевые слова: логарифмическое отображение Гаусса, особенности, дискриминант, амеба гиперповерхности, асимптотика.
