Mellin Transforms for Rational Functions with Quasi-elliptic Denominators Текст научной статьи по специальности «Математика»

EDN: HDYVVF УДК 517.55

Mellin Transforms for Rational Functions with Quasi-elliptic Denominators

Irina A. Antipova*

Siberian Federal University Krasnoyarsk, Russian Federation

Timofey A. Efimov^

MAEI Gymnasium no. 10 Divnogorsk, Krasnoyarsk Krai, Russian Federation

Avgust K. Tsikh*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 14.12.2022, received in revised form 04.09.2023, accepted 04.10.2023 Abstract. The paper deals with residue representations of n-dimensional Mellin transforms for rational functions with quasi-elliptic denominators. These representations are given by integrals over (n — 1)-dimensional relative cycles. The amount of representations (or cycles) equals to the number of facets of the Newton polytope for the denominator of the rational function.

Keywords: multidimensional Mellin transform, quasi-elliptic polynomial, Leray residue form, amoeba.

Citation: I.A. Antipova, T,A.Efimov, A.K. Tsikh, Mellin Transforms for Rational Functions with Quasi-elliptic Denominators, J. Sib. Fed. Univ. Math. Phys., 2023, 16(6), 738-750. EDN: HDYVVF.

Introduction

The fundamental property of the Mellin transform, which largely determines the scope of its applications, is the correspondence between the asymptotic behavior of the original function g(x) and the singularities of the transformed function. The role of this fundamental correspondence for the Mellin transform of a function of one variable is noted in numerous papers by F. Flajolet, in particular, in [4] in relation with the calculation of harmonic sums.

We recall that the Mellin transform of the function g(x) is defined by the integral

M [g](z)= f g(x)xz-1 dx, (1)

J R+

where the differential form

cz 1 dx := xZ1 ■ ... ■ x—1 —-Л ... Л

^d^c i dx—

— A ... A —-

x- x—

acts as a kernel. Inversion formulae for multidimensional Mellin transforms and classes of holo-morphic functions that can be translated into each other by direct and inverse Mellin transforms are studied in [1]. In this paper we deal with the Mellin transform of rational functions with

* iantipova@sfu-kras.ru https://orcid.org/0000-0003-1382-0799 ttimofeyefimov@yandex.ru

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

quasi-elliptic denominators f. Due to the specifics of the kernel, it suffices to consider the transform of the function g(x) = 1/f (x). The concept of a quasi-elliptic polynomial was introduced by T. Ermolaeva and A. Tsikh in [3].

So, we consider the following polynomial in n variables

f (x) f (x1, • • • : xn) ^ ^ Cax ^ ^ ca\...an xa • • • xa (2)

aeA aeA

with coefficients ca e C \ {0} and the support A c Z" .

Definition 1. A polynomial f is called to be quasi-elliptic if for any non-zero covector a e Rn* its truncation fa does not vanish in the torus (R\0)n.

Recall that the truncation of the polynomial f in the direction a e Rn* is determined to be the polynomial

E

aeAa

fa / cax ,

where Aa is the face of the Newton polytope of f in the direction a. The Newton polytope Nf of a polynomial f is defined to be the convex hull in Rn of the support A of f.

Meromorphic continuations of the Mellin transforms for rational functions with quasi-elliptic denominators were studied by L. Nilsson and M. Passare in [9], where the following representation

N

M[1/f](z) = $(z) H r (v(k) - ^(k),z)) , (3)

N

/ (k) / (k) z

k=l

was proved. Here $(z) is an entire function, vectors ¡j(k) £ Zn are primitive and define outward normal directions to facets of the Newton polytope Nf, and Z. The Newton polytope can

be given by the system of inequalities

N

•¡¡n

Nf = p| {u £ Rn : ^(k),u^ < v(k^,

L V 'U '

k=1

so each v(k) is interpreted as the weighted power of the polynomial f with respect to the corresponding weight ¡j(k). As it follows from the results of [3], the Mellin transform M[1/f](z) is a holomorphic function in the tube domain over the interior of the Newton polytope Nf, and formula (3) reveals that its polar set is the finite set of families of parallel hyperplanes. In each family, the hyperplanes are obtained by shifting of some facet of the Newton polytope of the polynomial f.

This approach was generalized for Euler-Mellin integrals in [2], and also found application in the theory of Feynman integrals, see [7,8].

In this paper, we present alternative representations for the Mellin transform of rational functions of the specified class. Note that we can define the quasi-ellipticity concept on the set R+ by assuming that truncations fa do not vanish on it, because the R+, being a connected component of the real torus (R\0)n, is its subgroup under the operation of coordinatewise multiplication.

Theorem 1. Let us assume that the polynomial f is quasi-elliptic on R+. Then for each normal vector ¡(k) of the Newton polytope Nf there is a representation for the Mellin transform M[1/f](z) of the following form

Mk (z) = e-in/^k)^r(-{^k),z))r(1 + ¡(k),z))$k (z), (4)

in which

$k (z) = v.p. Res u, (5)

JVk

where Res w is the Leray residue form of the integrand in (1), Vk is a surface of real dimension n — 1, and v.p. denotes the principal value with respect to the set of singular points of the Vk.

The function &k(z) defined by the integral (5) is holomorphic in the tube domain U^k] + IR", where

Uk] = fl {u G R" : (j)} .

j=k

1. Quasi-ellipticity and hypoellipticity

In this section, we characterize quasi-elliptic polynomials in more detail. First, note that the polytope Nf has only a finite number of faces, so the condition in Definition 1 needs to be verified only for a finite number of truncations fa.

Following [3], we can single out two classes of polynomials that are quasi-elliptic in the sense of Definition 1. The first class consists of polynomials in which all monomials have positive coefficients and even powers a in each variable xj. The second class consists of elliptic polynomials that do not vanish on R". Recall that a polynomial f is called elliptic if its homogeneous polynomial of highest degree vanishes in R" only at the point x = 0. Let us consider a few examples.

1. The polynomial

f (x) = 1 + 2xi + 2x2 + (xi — x2 )2 is not quasi-elliptic in R+, because its truncation

f(1,1) = (x1 — x2)2

vanishes on the diagonal x1 = x2.

2. The polynomial f =1 x1 ++ x 1 — x2 + x"2 — x1x2 is quasi-elliptic in R2.

3. The polynomial f = 1 + x1 + x2 is quasi-elliptic in R+ but not quasi-elliptic in R2.

The quasi-ellipticity property is related to the concept of hypoellipticity. A polynomial f is said to be hypoelliptic if for any multi-index a = 0 the derivative f(a) (x) satisfies the condition

f(a)(x) f (x)

for || x to [6]. The following sufficient test for hypoellipticity holds.

Theorem 2 (E. Zubchenkova). If f is a quasi-elliptic polynomial and its Newton polytope is full, then f is hypoelliptic polynomial.

Regarding the convergence of A-hypergeometric integrals, see articles [10] and [11]. The fullness of the polytope means that its projections on all coordinate planes belong to it. This condition in Theorem 2 is essential, as the following example confirms. The polynomial f (x1,x2) = x^x"2 + x\ + 1 is quasi-elliptic, but its Newton polytope is not full. The hypoellipticity condition is not satisfied for it, since for a = (4,0)

f(a)(x)

f (x)

= 24 ^ 0.

=0

We note also that the hypoellipticity condition does not imply the quasi-ellipticity one. For instance, the polynomial f (x1 ,x2) = (x2 — 1)2 + x2 is elliptic, and therefore hypoelliptic, but it is not quasi-elliptic, since the truncation f(o,-1) = (x2 — 1)2 has zeros in the torus (R\0)2.

Following [3], we now formulate the condition for the convergence of the integral of a rational function over R" with a quasi-elliptic denominator .

Theorem 3 (Ermolaeva-Tsikh). If Q is a quasi-elliptic polynomial non-vanishing in Rn, then the integral

i P (xi, ...,xn) , ,

I . , dx1 ... dxn

Q(x1 T ■ ■ ■ , xn )

is absolutly convergent if and only if

I + Np c (Nq)°,

that is, the translation of NP by I = (1,..., 1) e Rn lies in the interior (Nq)° of Nq. Regarding the convergence of A-hypergeometric integrals, see articles [10] and [11].

2. Sets Vk

The ortant R+ is a group with respect to the operation of coordinatewise multiplication. This is a connected component of the real torus (R\0)n. Any torus (R\0)n automorphism , as well as an automorphism of the R+, is defined by a monomial transformation

y ^ x = yn = (yni ,...,ynn),

where n1,... are rows of some integer unimodular matrix n (detn = ±1). The automorphism allows to integrate over R+ with fibers on shifts of one-parameter subgroups in R+.

Let us define the construction of sets Vk. For each outward normal ¡(k) of the Newton polytope Nf of the polynomial (2), we define a one-parameter subgroup

Yk = { yf := (yf' ) e R+ : yi e R+ j .

Next, we foliate the orthant R+ into shifts (cosets with respect to the subgroup Y(k)) as follows

c © Y(k) = (cy,...,cny^).

The set of all shifts can be given as c = (y')n, where y' := (y2,... ,yn), n' is an integer (n x (n — 1))-matrix such that n := (¡(k),n') is a unimodular (n x n)-matrix. The existence of such a matrix is ensured by the condition the vector ¡(k) to be primitive [13, Prop. 4.2.13].

Consider a section of the complex hypersurface V := {x e Cn : f(x1,...,xn) = 0} by a family of shifts of the subgroup 7(k). As a result, we get the set

^k = U (V n {x = y1<fc) © (y')"'T})

-1

of the real dimension n-1. This observation allows us to apply Fubini's theorem doing integration over arbitrary one-parameter fibers.

Let us describe this construction using the example of a complex hyperplane

0}

V = {x £ (C \ 0)2 : 1 + X! + x2 = 0} .

The Newton polytope of the defining polynomial is a triangle, it has outward normals ¡(1) = ( — 1,0), ¡(2) = (0, —1), ¡(3) = (1,1) (Fig. 1 on the left). Fig. 2 shows the real part of V and its sections by shifts of the one-parameter subgroups (1) (red ray), <2) (green ray )

(3)

and yM (blue segment). These are the sets V1, V2, V3 respectively. Their logarithmic images

y

are connected components of the contour of the amoeba AV of the hyperplane V (see Fig. 1 on the right). Recall that the amoeba of an algebraic hypersurface V is defined to be its image under the mapping

Log : (xi, ...,xn) ^ (log|xi|,..., log|x„|),

see, for example, [5]. The contour of the amoeba is determined as the set of critical values of the specified projection Log|V, i.e. the set Log(Y-1 (MP"-1)), where y : V ^ CP"-1 is the logarithmic Gauss mapping [12].

3. Proof of Theorem 1

Consider a polynomial (2) that has no multiple irreducible factors, i.e. df ^ 0 on each irreducible component of V. According to (1), the Mellin transform of the function 1/f is expressed by the integral

/xz-I

fx dx.

R+

Fix a normal vector ¡j,(k) = (p[k),..., fj,"-1), k = 1, . . . , N, of the polytope Nf. Let us

construct an integer unimodular matrix n in which the vector ¡j,(k) is the first column:

n(1) n(2) ■ ■ ■ n(n

(k) Vi (2) ni ■■ (n .n1

■ (k) Vn n ■ nn.. ■ ■ {n ■ nn

The monomial transform x = yn with coordinates

n12)

xi = Vi V2

(n)

V1 ■Vn ,

Xn = yin y2n ■■■ynn

is an automorphism of R+ due to the unimodularity of the matrix n- Let us write in variables y

dx

the expression —- The result looks as follows x

where

J

— = J]j (Vlj ■■■vl° ' )-1dyi A^^A dvn X j=i

., (k)

(n)

,.nj Vl.

rn./k' ,.ni2)-i yn

■ ■ ■ yn

(k) .nk)-1 Vn

Vn 'Vln ■■■Vnn

ni Vl V2

n (2)Ak\v(n2)-i

nn Vi V2

V(n) ■■■Vn

(n) .ik) nin)-i

ni 'Vi1 ■■■Vn1

(n)

(k)

nn Vi . . . Vn

lin)- i

is the Jacobian of the monomial mapping- Multiplying the jth row of the Jacobian by

(k) (2) .j Vj

(n)

i

Vi V2 ■■■Vn

we get the representation

dx

X

(k) -i (2) -i Vi V- ni V—

(k) -1 (2) -1 Vn 'Vi nn V-

(n)1 n1 Vni 1

(n)1 nn Vni 1

dVi A ■ ■ ■ A dVn ■

Further, from the j-th column of the determinant we take out the multiplier y- i- As a result, we get

dx dy

— = det n—, xy

where detn = ±1, i- e. the matrix n is unimodular.

As a result of the change of variables, the Mellin transform is represented by the integral

M[1/f (vv)](*)= det n

1 .(k),z) (n(2),z) (n(n),z) dV

1+ f(VV)

V1

V2

(n),

■ ■■ Vn ' ' —, V

where

f( V) (.(k)'a) (V(2)'a) (V(n)'a)

f (vv )=2-^ aaVi v2 ■■■Vn '■

aeA

Remark that maxaeA{(^(k), a}} = v(k), and this quantity is the degree of f (yn) over yi- Next, we integrate over yi for the fixed value y' = (y2, ■ ■ ■ ,yn) £ R+

l+-i:

M [1/f (vv )](z) = det n v2 ' ' ■■■ Vn " ^

(n(2),z) {V(n),z) dV

V1

dV1

i+-1

V' Jo f (VV) Vi '

n

Fig. 3. The integration contour r

In order to calculate the inner integral over y1, we introduce the complex variable £, setting y1 = Re£. Traditionally, we consider the integral over the contour r (see Fig. 3) of the following type

r ^\z) ^ = f_2MUWz)lAf ^(k)'z)-1 f ^(k)'z)-1 f &(k)'z)-1 If (&y')n) e =V )J f ((S,y')n) de f ((S,y')n) de + 7 f ((S,y')n) de

P Cr CP

Since the degree of f(yn) by y1 equals v(k), then, by the residue total sum theorem, this integral vanishes if (p(k), Res) < v(k). Passing to the limit as p ^ 0, R ^ <x and applying the residue theorem, we obtain

v{/k)'z) dyi

2ni

f (yn ) yi 1 _ e2ni{{^(k),z)-D j f (g (y'),y') '

E

g (y ' )

(Sk),z)-i

where g3 (y' ) are roots of f (yn ). Thus we get

M [1/f (yn )] =

2ni

1 _ e2ni{(^k),z)-i)

E

g (y' )

-1 3

2ni

f'i g (y'),y')

( xz-I

~yi

(n(2),z) (n(n),z) dy'

■■■yn

Res ( ^^-dx) ■ JVk Vf (x) J

1 _ e2ni((^k),z)-1) The factor before the integral can be rewritten as follows

2ni r(_{^(k),z))r(1 + (p(k),z))

1 _ e2ni{(^k),z)-i)

ei

T{(ß(V,z))

Thus, we obtain the first assertion of Theorem 1. The second one follows from the fact that the integral over Vk vanishes if (p(k), Rez) = (p(k),u) < v(k).

4. Examples

I. Consider the quasi-elliptic polynomial f (x) = 2 _ x1 + xl _ x2 + x2, _ x1x2 and the Mellin transform

u

y

R

+

M [1/f](z)= --+ f1*2 + 2-(6)

Jr2 - — Xl + xl — X2 + x2 — X1X2 x

The Newton polytope of the polynomial f (x) is given by three inequalities:

Nf = {_Z1 < 0} n {_Z2 < 0} n {Z1 + Z2 < 2}.

It has outward normals p(1) = (_1,0), p(2) = (0, _1), p(3) = (1,1). We consider the monomial change of variables

x1 = t-1, x2 = T-1,

associated with the vector ¡j(1) = (_1,0). As a result, the integral (6) will take the form:

[■ t1-zi T 1-z2

M [1/f](z) =JK 2t2T 2 _ tT 2 + T 2 _ t2 T + t2 _ tT dt A dT (7)

The denominator of the integrand in (7) has roots

(1) T2 + T + iT^7t2 _ 6t + 3 d 2) T2 + T _ iTsj7t2 _ 6t + 3 t = ^(It^t+I) and t = ^(It^t+I) .

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

M1(z) = r(z1)r(1 _ z1)einzi i Res w,

jvi

where

V1 = {x1 = 1/t(1), x2 = 1/t, t e [0;+to]} U{x1 = 1/t(2), x2 = 1/t, t G [0;+to]}.

Since the hypersurface V = {f(x) = 0} is smooth, the principal value v.p. in the representation is omitted. Thus,

M1(z) = r(z1)r(1 _ zx)einzi 2zi-1x

f™ (t2 + t + itv7t2 _ 6t + 3)1-zi _ (t2 + t _ itv7t2 _ 6t + 3)1-zi )

x / h dT. (8)

Jo Tz2 (2t2 _ T +1) 1 w7t2 _ 6t + 3

Now let us define the domain of convergence of the integral in (8). At the origin, its convergence is ensured by the condition

u1 + u2 < 2,

where u1 = Rez1, u2 = Rez2. Next, we study the convergence in the neighborhood of the infinity using the substitution t = 1/X. As a result, we obtain the integral

f™ (1 + A + iV7 _ 6X + 3X2) 1-zi _ (1 + A _ iV7 _ 6X + 3X2) 1-zi / -i-, --dX.

Jo X1-z2 (2 _ X + X2)1-zi w7 _ 6X + 3X2

The convergence of this integral at the origin is ensured by the condition u2 > 0. Thus, the integral on the right side of (8) converges in the tube domain with the base U^ = = {u G M2 : u1 + u2 < 2, u2 > 0}.

Next, consider the monomial change of coordinates

x1 = t, x2 = t-1,

associated with the vector ¡j,(2) = (0, —1)- As a result of this change of variables, the Mellin transform is expressed by the integral

r ti~Z2 TZl~i

M[1/f] = -^-^-dt A dT,

1 /JI JK2+ 2t2 — Tt2 + t 2t2 — t + 1 — tT '

in which the denominator of the integrand has roots:

f{i) = t + 1 + i^3t2 — 6t + 7 t{2) = t +1 — »v3t 2 — 6t + 7 = 2(t 2 — t + 2) , = 2(t 2 — t + 2) ■

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

M2(z) = r(z2)r(1 - Z2)einz2 i Resu,

JV2

ly2 where

V2 = {xi = t, x2 = 1/t(i), t £ [0; +to]} U {xi = t, x2 = 1/t(2), t £ [0; Thus, we obtain the formula

M2(z)=r(z2 )r(1 — Z2)einZ2 2Z2-ix

(t +1 + w3t2 - 6t + 7)1 z2 - (t + 1 - iV3r2 - 6t + 7)i z2

x ----—^-, ----dT. (9)

Jo ti-zi (t2 — t + 2) Z2 W3t2 — 6t + 7

The integral on the right side of (9) converges in a tube domain with the base U[2] = = {u £ R2 : ui > 0, ui + u2 < 2}-

Finally, consider the third normal ¡j(3) = (1,1) and the corresponding monomial mapping:

xi = t, x2 = Tt■

The Mellin transform takes the form:

f tZl+Z2-iTZ2-i

M [1/f](z) = l+ t2 (t 2 — T + 1)+ t ( —1 — t ) + 2 dt A dT The denominator of the integrand in the resulting integral has two roots:

t(i) = T + 1 + »V7T2 — 10t + 7 t(2) = T +1 — »V7T2 — 10t + 7 = 2(t 2 — t +1) , = 2(t 2 — t +1) ■

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

M3(z) = r(—zi — z2)r(zi + z2 + 1)e-in(Zl+Z2) i Resw,

Jv3

where

V3 = {xi = t(i), x2 = Tt(i), t £ [0; +to]} U {xi = t(2\ x2 = Tt(2\ t £ [0; Calculating the residue, we get the representation

Ms(z) = r(— zi — z2)r(zi + z2 + 1)e-in(Zl+Z2)2i-Zl-Z2 x

f~ (t +1 + iV7T2 — 10t + 7)Zl+Z2-i — (t + 1 — iV7T2 — 10t + 7) Zl +Z2-i ( )

Fig. 4. Contour of the amoeba for f (x) = 2—x1+x\ —x2+x\-xLx2 and Log(VL),Log(V2),Log(V3)

The integral in (10) converges in a tube domain with the base U[3] = R+.

In Fig. 4, the logarithmic projections of the sets Vl, V2, and V3 are shown in blue, green, and yellow, respectively. The contour of the amoeba is highlighted in red.

II. Consider a quasi-elliptic polynomial f (x) = 5 + xL + x2 + xLx2 and the Mellin transform:

M [1/f ](z)

b1 x2

dx

5 + xi + X2 + X1X2 x

(11)

The Newton polytope of the polynomial f (x) is given by the inequalities

Nf = {— zL < 0} l~l {— z2 < 0} n {zL < 1} l~l {z2 < 1} and therefore has outward normals = ( —1,0), ¡j(2) = (0, —1), ¡j(3) = (1,0), ¡j(4) = (0,1).

We consider the monomial change of variables

xi = t

1

x2 = T

1

associated with the vector ¡(1) = ( —1,0). The Mellin transform after the change of variables will take the form:

M [l/f](z) =

t

-Zi —Z2

R+ 5tT +1 + T +1

-dt A dT.

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

where

Mi(z) =r(zi)r(1 - zi)einZiv.p. i Resu,

J Vi

5T + 1 1

Vi ^ xi = - t + 1 ,x2 = t—i,T G [0;+œU .

Calculating the residue, we get the following result:

Mi(z) = r(zi)r(1 — zi)e^ / ( —1)"

!(t + 1)-

(5t +1)

i —zi

dT.

(12)

u

The integral in (12) converges in the domain {(z1,z2) G C2 : 0 < Rez2 < 1}}. Next, consider the monomial change of coordinates

X1 = T, X2 = t—1,

associated with the vector ¡j(2) = (0, —1). As a result of this change of variables, the Mellin transform is expressed by the integral

f TZi—1t~Z2

M [1/f ](z) = + t + dt A dT.

J r+ 5t + Tt + T + 1

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

M2(z) = r(z2)r(1 — z2)einz2v.p. i Resw,

Jv2

where

V = jX1 = T,X2 = — T+5,T G [0; +œ] j . Calculating the residue, we obtain the result:

~ TZl —1(t + 1)—Z2

(t + 1)-

M2(z) = r(z2)r(1 - Z2)einz2 (-1)-z2 V '-z2 dT. (13)

Jo (5 + t)

The integral in (13) converges in the domain {(zi,z2) £ C2 : 0 < Rezi < 1}}. Further we consider the vector v(3) = (1,0) and do the substitution

Xi = t, X2 = T■

As a result, the Mellin transform takes the form

r tzi-i Tz2-i M [1/f](z)= + +t + t dt A dT. jv2+ 5 + t + t + tT

According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

Ms(z) = r(-zi)r(1 + zi)e-inziv.p. f Resu,

jv3

where

= ^xi = TT+ 15, X2 = t,t £ [0; j . Calculating the residue, we get the representation

rTz2-i(T+5)zi-i

M3(z) = r(-zi)r(1 + zi )e-inzi (-1)zi-i-^-dT. (14)

jo (1+t)

The integral in (14) converges in the domain {(zi,z2) £ C2 : 0 < Rez2 < 1}}.

Finally, we consider the normal v(4) = (0,1) and the corresponding monomial mapping:

Xi = T-i, X2 = t■

The Mellin transform after the change of variables is as follows:

f T-zi tz2-i M [1/f]= 5 +.+t + t dt A dT. JR2+ 5t + 1 +1 + tT

6 -

m

I

2 -

-4 -3- -2 -1 0

-2 -

-4-

Fig. 5. Contour of the amoeba of V = {5 + x\ + x2 + x\x2 = 0} and logarithmic projections of Vk According to Theorem 1, the Mellin transform M[1/f](z) admits the representation

The integral in (15) converges in the domain {(z1,z2) £ C2 : 0 < Rezi < 1}. The contour of the amoeba of V = {5 + x1 + x2 + x1x2 = 0} is shown in Fig. 5. The sets Log(V1) and Log(V3) coincide with the green part of the contour, and the sets Log(V2) and Log(V4) coincide with the yellow one.

The research is supported by the Russian Science Foundation, project no. 20-11-20117.

References

[1] I.A.Antipova, Inversion of many-dimensional Mellin transforms and solutions of algebraic equations, Sb. Math., 198(2007), no. 4, 447-463.

DOI: 10.1070/SM2007v198n04ABEH003844

[2] C.Berkesch, J.Forsgard, M.Passare, Euler-Mellin Integrals and A-Hypergeometric Functions. Michigan Math. J. 63 (2014), 101-123.

[3] T.O.Ermolaeva, A.K.Tsikh, Integration of rational functions over 1" by means of toric compactifications and multidimensional residues, Sb. Math., 187(1996), no. 9, 1301-1318. DOI: 10.1070/SM1996v187n09ABEH000157

where

Calculating the residue, we get the representation

(15)

[4] P.Flajolet, X.Gourdon, P.Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoret. Comput. Sci., 144(1995), 3-58. DOI: 10.1016/0304-3975(95)00002-E

[5] M.Forsberg, M.Passare, A.Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. Math., 151(2000), 45-70. DOI: 10.1006/aima.1999.1856

[6] L.Hormander On the theory of general partial differential operators, Acta Math., 94(1955), 161-248.

[7] R.P.Klausen, Hypergeometric series representations of Feynman integrals by GKZ hyperge-ometric systems. J. High Energ. Phys., 2020(2020), 121. DOI: 10.1007/JHEP04(2020)121

[8] R.P.Klausen, Kinematic singularities of Feynman integrals and principal A-determinants, J. High Energ. Phys., 2022(2022), 4. DOI: 10.1007/JHEP02(2022)004

[9] L.Nilsson, M.Passare, Mellin transform of multivariate rational functions, Journal of Geometric Analysis, 23(2013), 24-46. DOI: 10.1007/s12220-011-9235-7

[10] L.Nilsson, M.Passare, A.K.Tsikh, Domains of convergence for A-hypergeometric series and integrals, J. Sib. Fed. Univ. Math. Phys., 12(2019), no. 4, 509-529.

DOI: 10.17516/1997-1397-2019-12-4-509-529

[11] M.Passare, A.K.Tsikh, O.N.Zhdanov, A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals, Aspects Math. E, 26(1994), 233-241.

[12] M.Passare, A.Tsikh, Amoebas: Their spines and their contours, Contemporary Math., 377(2005), 275-288. DOI: 10.1090/conm/377

[13] T.M.Sadykov, A.K.Tsikh, Gipergeometricheskie i algebraicheskie funktsii mnogikh peremen-nykh, Nauka, Moscow, 2014 (in Russian).

Преобразование Меллина для рациональных функций с квазиэллиптическими знаменателями

Ирина А. Антипова

Сибирский федеральный университет Красноярск, Российская Федерация

Тимофей А. Ефимов

МАОУ Гимназия № 10 Дивногорск, Красноярский край, Российская Федерация

Август К. Цих

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В статье рассматриваются вычетные представления те-мерных преобразований Меллина для рациональных функций с квазиэллиптическим знаменателем. Эти представления задаются интегралами по (п — 1)-мерным относительным циклам. Количество представлений (или циклов) равно числу граней многогранника Ньютона знаменателя рациональной функции.

Ключевые слова: многомерное преобразование Меллина, квазиэллиптический полином, форма-вычет Лере, амёба.