On the Non-standard Interpolations in Cn and Combinatorial Coefficients for Weil Polyhedra Текст научной статьи по специальности «Математика»

EDN: HRVUPS УДК 517.55

On the Non-standard Interpolations in Cn and Combinatorial Coefficients for Weil Polyhedra

Matvey E. Durakov*

Siberian Federal University Krasnoyarsk, Russian Federation

Roman V. Ulvert^

Siberian Federal University Krasnoyarsk, Russian Federation Reshetnev Siberian State University of Science and Technology

Krasnoyarsk, Russian Federation

August K. Tsikh*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.08.2023, received in revised form 27.09.2023, accepted 24.10.2023 Abstract. Multidimensional non-standard interpolation has been recently presented in an article by D. Alpay and A. Yger. We are talking about algebraic interpolation where discrete roots of a system of polynomial equations serve as nodes. With the help of the Grothendieck residue duality, the problem of describing the desired interpolation space of functions is reduced to solving the affine-bilinear equation. To implement this reduction, algorithms for calculating local Grothendieck residues or their sums are required. In a fairly general situation, the calculation of these residues is based on the well-known Gelfond-Khovanskii formula. This article provides examples of calculating local residues or their sums. In 2-dimensional case we generalise the Gelfond-Khovanskii formula for Newton polyhedra that are not in the unfolded position. This is done using the concept of an amoeba of an algebraic set and the notion of an homological resolvent for the boundary of Weil polyhedron. Keywords: Grothendieck residue, interpolation, amoeba, Homological resolvent

Citation: M.E. Durakov, R.V. Ulvert, A.K. Tsikh, On the Non-standard Interpolations in Cn and Combinatorial Coefficients for Weil Polyhedra, J. Sib. Fed. Univ. Math. Phys., 2023, 16(6), 758-772. EDN: HRVUPS.

Introduction

By classical, or standard, interpolations we understand the Lagrange, Hermite, or Newton interpolations. Let us consider the first two of them.

Problem (Lagrange). Given a set of distinct points {wjC C and the values Cj € C, find the polynomial f (z) of degree m — 1 with the property

f (Wj ) = Cj, j = !,...,m.

*durakov_m_1997@mail.ru tulvertrom@yandex.ru

£ atsikh@sfu-kras.ru https://orcid.org/0000-0002-2905-9167 © Siberian Federal University. All rights reserved

Note that the interpolation polynomial f is defined in terms of the polynomial p(z) = (z — w1) ■ ... ■ (z — wm) by the formula:

m c- (1

f (z) = p(z) V-j— res -

j=i

z — Wj Wj yp

(pp)

Thus, specifying the interpolation nodes in the form of the null set of the polynomial p provides tools for constructing an interpolation polynomial by using residues. More general is the following

Problem (Hermite). Let {w-}m=1 C C be a set of pairwise distinct points and the following values are given

Cji £ C, where j = 1,... ,m, l = 0,..., ¡j — 1.

It is necessary to find a polynomial f (z) of minimal degree, which at points w- has the given values of derivatives up to orders of ¡ij — 1 including, that is

f (l)(wj ) = Cji, j = 1,...,m, l = 0,...,ij — 1. (1)

In the Hermite interpolation problem, it is advisable to enumerate the set of points wj taking into account their multiplicities, thereby considering the set {wj}j=1 as an algebraic set p-1(0), where

p(z) = (z — wif1 ■ ... ■ (z — wm)Mm. (2)

The Hermite interpolation polynomial can be represented as

/ , p(z\ T j ( T (z — wj )i+s resf(z—wri

(z — wj)»j ^ l! \ ^ v j Wj

j ( £ (z — wj)

z - w-i )l+s res

j=1K J' 1=0 \ s=0

that is, again using residues.

Recently there have been papers on so-called non-standard interpolation (see [1,2]). They pose the problem of constructing a function f on an algebraic set p-1 (0) (considered as an analytical space), whose values lie on a given hypersurface. More precisely, the article [1] discusses the following:

Problem. Given the complex numbers aj,k (j = 1,...,m; k = 0,...,ij — 1) and c. It is necessary to describe the set of all functions f which are analytic in the neighborhood Q C C of points w1,..., wm and satisfy the equation:

m Mj — 1

f {k)(w0 ) = c. (3)

j = 1 k=0

Note that if f is a solution of (3), then f + ph is also a solution, where

m

p(z) = Y[(z — wj, h £O(Q).

j=1

In other words, we can work in the factor ring O(Q)/(p) by the ideal generated by the polynomial p.

In a similar form, a non-standard problem can be posed in the multidimensional case, treating the interpolation nodes w1,..., wm £ C" as the zeros of an ideal (p) = (p1(z),... ,pn(z)) in the ring of polynomials in the variable z from C". In the main part of the article, in particular, we will demonstrate a non-trivial example of such multidimensional interpolation.

1. Multidimensional non-standard interpolation

To formulate a multidimensional non-standard interpolation problem, we need the following definition of a Noetherian operator.

Definition 1 (Ehrenpreis [15], Palamodov [14]). Let I C C[si,...,s„] be a primary ideal. A family of linear differential operators with polynomial coefficients di(s, D), 1 = 1,... ,t is called a Noetherian operator for I, if the conditions

de(s,D)y(s)\v (i) =0 W =1,...,t

are necessary and sufficient for the function y(s) to belong to ideal I.

In the one-dimensional case an arbitrary polynomial has the form:

p(s) = (s - wif1 • ... • (s - wkyk,

and its generated ideal is decomposed into the intersection of primary ones

Pj = {(s - Wj Yj ), j = 1,...,k.

A necessary and sufficient condition for a given function y to belong to the primary component Pj is vanishing of y by the following operators with constant coefficients:

r / M T

where Cij [y(s)] = -y—

r r r

Lj,0, Lj,l, ■ ■ ■ , -1,

dj p

dsj .

s=-Wi

For an arbitrary n ^ 1, the primary components pj of a zero-dimensional polynomial ideal (p1. ,pn) are attributed to the roots Wj. The Noetherian operators pj are the arrangements of differential operators with constant coefficients.

Lwji(d/ds)\wj, I € Awj.

Here AWj is a finite subset in Nn

Now we can proceed to the formulation of the multidimensional non-standard interpolation problem.

Problem 1 ([2]). Let p-1(0) = {wi ,...,wm} and U be an open subset of Cn containing p-1(0). Fix aj,i, j = 1,... ,m, l € AWj and c; all of them are complex numbers. We need to describe the space of holomorphic functions f: U ^ C with the following property:

m

Y^ Y aj,icwoAf\(wj) = c (4)

j=i eeAwj

The following monomial basis

B = {s^k; k = 0,...,N(p) — 1}

in the quotient space C[z]/(p) is one of ingredients for solving the interpolation problem. In fact, this factor is the space of remainders when dividing polynomials by the ideal (p).

2. Grothendieck residue and its role in interpolation theory

The Grothendieck residue is a cornerstone of complex analysis and algebraic geometry and it plays the key role in the singularity theory and foliations theory. Assume that the sequence of germs

f1,...,fn £ C[z] = C[z1,...,zn] have an isolated common zero at a £ Cn. Consider a meromorphic differential n-form

1 h(z) dz ,--,1 n

W = (n)n f1(z) .. . fn (z) (with dz = dz1 dz").

Definition 2 ([4,5]). The Grothendieck residue, associated with f = (f1:...,fn) and h, is determined as an integral

res (h) = w

a f Jra

of the form w over a very special cycle

ra = {z £ Ua : \ fj (z)\ = Ej ,j = 1,...,n},

where the neighborhood Ua of a and e- are chosen such that the closure Ua does not contain roots different from a and ra CC Ua.

We call the integration set ra a Grothendieck cycle. Note that this is the skeleton of the Weil polyhedron {z £ Ua: \fj (z) \ < e- ,j = 1,...,n}.

In the case of a finite set of zeros, the mapping p = (p1,... ,pn) can define the global Grothendieck residue as the sum of the local ones. The global residue is denoted by h(s) ds1 A ■ ■ ■ A dsn

Res p).

p1(s),...,pn(s)

, h £ H(D) (D is the domain containing all zeros of the mapping

Now, using the notation introduced above, we can state a theorem that gives the way to solve Problem 1.

Theorem 2.1 (Alpay, Yger [2]). Let {w1,... ,wm} = p-1(0), U be an open subset in Cn containing p-1 (0). Let the sequence

a = {aj,i, j = 1,.. .,m, 1 £ Awj}

and the complex number c be given. Let us denote the polynomials

hWj (s)= T ajl(s — w-)*/£!,

leAwj

making up the sequence hW = [hW1,..., hWm], and let

a[hW] = ^ofr^i.. .,aN (p)-1[hW])

be the projection of this sequence onto the quotient space C[z]/(p).

• If a[hW] = 0, then the problem has no solution in the case c = 0, and any function f £ O(U) is a solution in the case c = 0;

• If a[h'Wu] = 0, then f £ O(U) satisfies the condition (4) iff

a[f] ■ Qp[B] ■ a[hW]T = c, where T is the transposition sign, and Qp[B] is the Grothendieck global residues matrix:

sfai +Pk2 ds

Qp[B] = Res

p1(s) . . .pn(s)

0<ki,k2<N <p>-1

3. Amoeba and its complement

For further reasoning we will need to introduce the concept of the amoeba of the Laurent polynomial, as well as describe some of its properties.

Definition 3. Given a Laurent polynomial f its amoeba Af is the image of the hypersurface V = f-1(0) under the map

Log: (zi,...,zn) — (log\zi\,..., log \zn\).

For the amoeba we will also use notation AV.

Amoeba reflects the distribution of the algebraic set V. More precisely, one can say that the amoeba depicts hollows for V.

The shape of the amoeba is closely related to the Newton polytope Af of the polynomial f. Recall that Af is defined as the convex hull in Rn of the index set A in the experession

f (Zh...,Zn) = Y^

a£Ä

The set of integer points in Af admits a natural partition Af n Zn = |Jr Ar, where r is a face on Af and Ar denotes the intersection of Zn with the relative interior of r. We shall consider the dual cone Cv of Af at v defined as

Cv = < s € Rn : (s,v) = max (s, .

I J

Notice that dim Cv = n — dim r when v € Ar. In particular, Cv has nonempty interior if v is a vertex of Af, and it equals {0} whenever v is an interior point of Af.

The following theorem allows us to introduce an order on the components of the amoeba complement.

Theorem 3.1 (Forsberg, Passare, Tsikh [13]). On the set {E} of connected components of cAf there is an injective map (the order map)

v: {E} — Af n Zn

with the property that the dual cone CV(E) is equal to the recession cone of E. That is, for any u € E one has u + Cv € E and no strictly larger cone is contained in E (Notice that if v is the k-skeleton of Af the Cv has dimension n — k).

Thus, connected components can be numbered as Ev with integer v e Af. See, for examples, the figures below for the polynomial 1 + z2z2 + ziz| + 5ziz2 (Fig. 1).

4. The Gelfond-Khovanskii formula

We say that the sequence of polytopes A1,..., An € Rn is unfolded, if for each covector v € (Rn)* there is such number i, that for vectors x € A = A1 + ... + An (the Minkovskii sum of polytopes Aj) the scalar product (x,v) take its maximal value only in some vertex of Aj.

£ resf (h) = £ kv Res( f h a E \n ■■■JnJ

Theorem 4.1 (Gelfond-Khovanskii [16]). Assume that the Newton polytopes Ai:..., An of polynomials fi,..., fn are unfolded. Then the sum of all local residues in (C \ 0)n is calculated by the formula:

h

{a} veVertA E ^fl . . . fn

h

where ResEv is the coefficient c-I of the Laurent decomposition for —-— in the connected

fi...Jn

component Ev.

In fact one can prove that the sum ^ ra of local Grothendieck cycles ra is homologically

{a}

equivalent to the sum

y^ kv Log-i(uv), uv e Ev,

veVertA

where kv are the combinatorial coefficients: We ascribe the combinatorial coefficient to each vertex v of the sum A of unfolded polytopes. Each face r c A is a sum r + ... + rn of faces r c Ai.

Definition 4 ( [16]). Combinatorial coefficient kA is the local degree of the germ

(dA,A) ^ (dR+, 0)

of the characteristic map (hi,..hn) : d A ^ dR+, where each component hi is zero precisely on that face of r, for which the term r is a vertex of Ai.

5. The homological resolvent

Let U = [Ui] be a finite covering of some manifold X. Denote SU the group generated by all singular simplices of dimension q which supports belong to some element Ui of the covering U. Let C*(U, S*) be the complex (which can be called the Cech-de Rham complex in homological version) formed by the group SU, that is bigraduated groups

Cpq = © Sq (Ui0 n Ui! n ... n Uip), p,q = 0, 1,...

io<ii<...<ip

We will need the following definition later to calculate local Grothendieck residues. The definition given below differs slightly from Gleason's definition [17].

Definition 5. The sequence of U-chains {Cp}p=o, Cp € Cp,r-p we will call the U-resolvent of the cycle C € Zr (SU) if the following two conditions are met:

1. C = c.

2. 5Cp = dCp-1, p = 1,...,r.

Here e : C0— SU is an inclusion operator Ui c X the action of which is determined by the formula of the alternated sum:

ea = 53 a(i).

iei

Boundary operator d : Cq —^ C* q— 1 is defined as

(da)(io, i1,..., ip) = d (a(io, h,..., ip)). The Cech coboundary operator: Inclusions

Ui0 n uh n... n Uiv — Ui0 n Uj! n... [ifc]... n Uiv, k = 0,...,p, induce the operator 5 : Cp^ — Cp-1^ determined using the alternated sum formula:

(Sa)(io, i1,..., ip-1) = ^2 a(i, io, .. ., ip-1). iei

6. A generalisation of the Gelfond-Khovanskii formula in 2-dimensional case

Let us consider an example of the polynomial system of equations:

F1 = 3z?z2 + z| + 2z1zf =0,

F2 = z3 + 4z1z3 + 3z2z2 = 0, (5)

for which the Newton polytopes are not in unfolded position: A1 and A2 have parallel edges. For convenience, we introduce the notation:

F1 = z2(3z2 + z23 + 2z1z2) = z2f1, F2 = z1 (z2 +4z2 + 3z1z2) = z1f2.

Amoebas of Af1 and Af2 are shown in Fig. 2. In the complement of amoeba Af1f2 there are 6 connected components: E51,E25,E17 corresponds to the vertices of A = A1 + A2, and E13, E26, E24 correspond to integer points in the relative interiors of the edges.

We will find the formula for the resolvent of the boundary dW of the Weil polyhedron W = Log-1(zA) , which is defined by a homothetic dilatation of the triangle A which contains the intersection of the amoebas A1 and A2. So, W contains all roots of the system F1 = F2 = 0 in the torus (C\{0})2. Note that the sets

U1 = {(C\{0})2} \{z : F1(z) = 0}, U2 = {(C\{0})2} \{z : F2(z) = 0}.

Fig. 2. Newton polytopes A1; A2, A and amoebas Af1, Af2

form a covering U of the complement (C\{0})2\{z : F1(z) = F2(z) = 0}. We want to construct the resolvent for the cycle C € Z2(SU) which is the boundary of the polyhedron W.

1st step: Decomposition C = a1 + a2 by blue and red chains with supports supp aj c Uj (see Fig. 3). Therefore we can take Co € Co,2 in Definition 5 as the following:

iCo(1) = a1, \Co(2) = a2

(Note that each support of aj consists of 2 connected components).

Fig. 3. The boundary of the Weil polyhedron.

2nd step: Computation of the boundary of chain Co gives

i(dCo)(1) = d(Co(1)) = da1 = r26 — r25 + ^3 — r24, \(dCo)(2) = d(Co(2)) = da2 = r24 — r26 + r25 — T43-

Now let ^ e C12 be our chain from the resolvent. Then by inclusions Ui n Uj ^ Ui we get the following system:

f №i)(1) = EieI £i(i, 1) = £i(1,1) + £i(2,1) = —£i(1, 2), l№)(2) = Ei£I £i(i, 2) = £i(1, 2) + £i(2, 2) = £i(1, 2).

Therefore if we take into account the fact that 6£1 should be equal to d£0 then from the system i —£i(1, 2) = (5£i)(1) = (d£o)(1) = r26 - ^5 + ^3 - r34,

\£i(1,2) = (S£i)(2) = (d£o)(2) = r34 — r26 + r^ — r43 we have the following expression for the resolvent:

£i(1, 2) = ^4 — r26 + ^5 — r43. Thus we get the following formula for the sum of the Grothendieck residues:

£ r<asp (h)=Res{ fF)—Res{ fF)+Res{ ff)— Res{ fF) ■

{a}

Obviously, the following statement in dimension 2 is obtained by similar reasoning.

Theorem 6.1. Assume that the system Fi = F2 =0 has in (C\[0])2 a finite number of roots. Then the sum of Grothendieck residues in the torus (C\{0})2 is calculated by the formula

Eresp (h) = kv Resf —| , where kv e [0,1, —1].

a Ev \F1F2

{a} veZ2nSA V 1 2

7. Example in dimension 3

Let us consider an example of non-standard interpolation when the single point a = 0 is defined as an isolated zero of the polynomial system

Pi = zi — Z2Z3 = 0, P2 = Z23 — Z1Z3 = 0,

P3 = Z3 — Z1Z2 = 0.

We have an open covering U for the punctured neighborhood U of the origin:

Ui = U\[z : Pi(Z) = 0], i = 1, 2, 3.

Now we want to construct the resolvent for a multicoloured cycle £ e Z5(S*U) (in the picture 4 below) which is homeomorphic to the sphere S5.

1st step: Decomposition £ = a1 + a2 + a3 by blue, red and green chains (see Fig. 4) with supports supp Oj c Uj. Therefore we can take £0 e C0,5 as the following chain:

[£o(1) = 01, £0(2) = 02, xio(3) = 03]

(Note that each support of Oj consists of 2 connected components).

0.0 x 0.0

Fig. 4. Hypersurfaces (left) and toric polyhedron (right) on the Reinhardt diagram

2nd step: Computation of the boundary of chain Co (see Fig. 5):

(dCo)(1) = d(Co(1)) = da1 = c + b — f — g — d + h, (dCo)(2) = d(Co(2)) = da2 = —a — c + e + d — i + g, (dCo)(3) = d(Co(3)) = da2 = —h — e + f + i — b + a.

Fig. 5. Orientation and indexing of chains

Now let G Ci,4 be our chain from the resolvent. Then by inclusions Ui n Uj ^ Ui we get the following system:

'№)(!) = E&&!) = !) + £i(2, !) + £i(3, !) = 2) " £i(1> 3),

iei

, (¿a)(2) = eei(i,2) = a(i,2) + ^2,2) + a(3,2) = a(i,2) - 3),

iei

(s^i)(3) = E £i(i> 3) = ei(i, 3) + ei(2,3) + a (3,3) = a(i, 3) + ^2,3).

iei

Note that chain C1(1, 2) can contain only green segments (d,c,g) [only these segments belongs to both sets U1 and U2], chain C1(1,3) can contain only red segments (b,f,h) chain C1(2,3) can contain only blue segments (a, e, i). Therefore if we take this fact into account together with the fact that 5C1 should be equal to dCo then from system:

—C1(1, 2) — C1(1, 3) = (5C1)(1) = (dCo)(1) = C + b — f — g — d + h, C1(1, 2) — C1 (2, 3) = (5C1)(2) = (dCo)(2) = —a — c + e + d — i + g, C1(1, 3) + C1 (2, 3) = (5C1)(3) = (dCo)(3) = —h — e + f + i — b + a.

we have the following solution:

C1(1, 2) = d — c + g,

C1(1, 3) = —b + f — h, C1(2, 3) = a — e + i.

3rd step: Compute the boundary of the chain C1 (see Fig. 6):

J (dC1)(1, 2) = d(C1(1, 2)) = d(d — c + g) = dd — dc + dg = — r + ro — r — r2,

J (dC1)(1,3) = d(C1(1,3)) = d(—b + f — h) = —db + df — dh = r + r — ro + r2,

I (dC1)(2, 3) = d(C1 (2, 3)) = d(a — e + i) = da — de + di = — r + ro — ^ —

Fig. 6. 4-dimensional chains a,b,... (left) and toric cycles ro, ri,... (right)

Now let £2 G C2 3 be our chain from the resolvent. Then by inclusions U1 n U2 0 U3 ^ Ui n Uj we get the following system:

(5£2)(1, 2) = E1, 2) = £2(1,1, 2) + £2(2,1, 2) + £2(3,1, 2) = £2(1, 2, 3),

iei

№)(1, 3) = E£2(i, 1, 3) = £2(1,1,3) + £2(2,1,3) + £2(3,1,3) = -£2(1,2,3),

iei

(S£2)(2, 3) = E£2(i, 2, 3) = £2(1, 2, 3) + £2(2, 2, 3) + £2(3, 2, 3) = £2(1, 2, 3). iei

Therefore by resolvent condition we have:

'£2(1, 2, 3) = (S£2)(1, 2) = (d£ 1 )(1, 2) = —^ +ro — r 1 — r3, —£2(1,2,3) = (S£2)(1, 3) = (d£i )(1, 3) = r i+r3 — ro + r2, £2(1, 2, 3) = (S£2)(2, 3) = (d£i)(2, 3) = + ^ — r — r.

That is why £2(1, 2, 3) = ro — r — r2 — r3, and therefore the Grothendieck cycle admits a representation

r = r222 — r511 — ri51 — ri15

by toric cycles. This fact helps us to construct the matrix Qp[B] from Theorem 2.1. For example, let us compute the following integral.

1 f cZfZy zY dz c f Z1Z2 zY dZi dZ2 dZ3

(2ni)3 J P1P2P3 (2ni)3 J (z3 — Z2Z3)(Z3 — ziz3)(z$ — Z1Z2) r 222 T222

Z1Z2 z3 dzi dz2 dz3

(2ni)3r{22 (—Z2Z3)(—Z1Z3)(—Z1Z2)(1 — zZfcX1 — zZtX1 — zZt)

z'a 2z2, 2z3Y 2 dzi dz2 dz3

(2ni)3 J (1__zL)(1__SL)(1__zl_)

r222 V Z2Z3/V Z1Z3/V Z1Z2'

/ z3 \m / z3 \n / z3 \l

£SS(£-2zY -2dz

<2nW J ^^ — } } \ ~ }Zi Z2 Z3

v > J m=o n=o l=o r 222

= —cY —L- f z3m-n-l+a-2zln-m-l+l3-2z\l-m-n+Y-2 d,Z = —Ck.

^ (2ni)3 J 1 2 3

m,n,l>o v ' p

V 222

Here k is the number of sets of non-negative integers [m, n, l] for which the following system of equalities holds:

3m — n — l + a — 2 = —1, —m + 3n — l + ¡3 — 2= —1, —m — n + 3l + y — 2 = —1.

Let us consider an example of non-standard interpolation when the single point a = 0 is defined as an isolated zero of the polynomial system

P1 = z3 — Z2Z3 = 0, P2 = z\ — Z1Z3 = 0,

P3 = Z3 — Z1Z2 = 0.

The multiplicity at 0 equals 11.

The Grothendieck cycle admits a representation

ro = r222 — r511 — r151 — r115.

c

c

c

Proposition 1. The list of Noetherian operators for the ideal I0 (P) is:

=

L0,100 = L0,001 = L0,011 = L0,002 = Lo,030 = ^0,040 =

|L0,000 = ^

= -d0-

d3

1 d4

dz1dz2dz3 ; L0,010 =

; L0,110 =

4! dz4 4! dz4

I _di _ d2

3! dz3 dz2dz3

_!_di _ d2

3! dzf dz1dz2

■ ^);£0'200 =(" 4 ;

-1 ; C, = ( d0

1

4! dzf"

I _ d2

3! dz3 dz1dz3 d

1 d4

4! dz4

020 =

(" 4 dz2)

= f — 1 ^ ;

(— 4 §4 ); L°'m = ("d0); L0'300 = (

); L0,400 = (-4"!d0) ;

1 d 3! dz1

IA

"3!

■1 d0 4!

; £0,003

^0,004 =

I_d_

3! d^

■1 d0 4!

Proposition 2. The monomial basis for the factor-space O0/I0(P) is

222

{1, z1, z2, z3, z1, z2, z3, z1z2, z1z3, z2z3, z1z2z3}.

Now we can formulate the local non-standard interpolation problem and its solution.

Problem 2. Let the complex numbers {ae}eeAa and c be given. Let U0 is an open subset of C3, containing the point 0 = (0, 0,0) which is an isolated zero of mapping P = (z3 — z2z3, z3 — z1z3, zf — z1z2). It is necessary to describe the space of holomorphic functions f : U0 ^ C, with the property:

]T aeC0/[f](0) = c.

eeAo

Theorem 7.1. If a[hW] = 0, then the holomorphic function f (s) satisfies the Alpay-Yger problem for single point (m = 1) iff the coordinatization of f satisfies the following condition:

a000 + am —

(

I ( I a030) m , ( , a003) m a200 r J.-1 . + ^101 + J a3 [f ] + ^110 + j a4 [f ] + a5 [f ]+

a400 + a040 + a004 \ r,n , / , 0,300 \ r,n , a1[f ] + (^011 + ) a2 [f ] +

+ "Tp a&[f ] + "Tp ar[f] + a001ag[/] + a010ag[f] + a001a10[f] + a000an[f] = —c.

This means that the coordinate vector of f in the local algebra lies in the prescribed affine hyperplane na c C11.

The investigation was supported by the Russian Science Foundation, grant no. 20-11-20117.

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О нестандартных интерполяциях в Сп и комбинаторных коэффициентах для многогранников Вейля

Матвей Е. Дураков

Сибирский федеральный университет Красноярск, Российская Федерация

Роман В. Ульверт

Сибирский федеральный университет Красноярск, Российская Федерация Институт информатики и телекоммуникаций Сибирский государственный университет науки и технологий им. М. Ф. Решетнева

Красноярск, Российская Федерация

Август К. Цих

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Многомерная нестандартная интерполяция была недавно представлена в статье Д. Алпая и А. Ижера. Речь идет об алгебраической интерполяции, в которой узлами служат дискретные корни системы полиномиальных уравнений. С помощью двойственности вычета Гротенди-ка задача описания искомого интерполяционного пространства функций редуцируется к решению афинно-билинейного уравнения. Для реализации этой редукции требуются алгоритмы вычисления локальных вычетов Гротендика или их сумм. В достаточно общей ситуации вычисление указанных вычетов основано на известной формуле Гельфонд-Хованского. В данной статье приведены примеры вычисления локальных вычетов или их сумм. В двумерном случае мы обобщаем формулу Гельфонд-Хованского для многогранников Ньютона, которые не находятся в развернутом положении. Это делается с использованием понятия амебы алгебраического множества и понятия гомологической резольвенты для границы многогранника Вейля.

Ключевые слова: вычет Гротендика, интерполяция, амёба, гомологическая резольвента.