A LIST OF INTEGRAL REPRESENTATIONS FOR DIAGONALS OF POWER SERIES OF RATIONAL FUNCTIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-5-624-631 УДК 517.55

A List of Integral Representations for Diagonals of Power Series of Rational Functions

Artem V. Senashov*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 29.03.2021, received in revised form 16.04.2021, accepted 20.06.2021 Abstract. In this paper we present integral representations for the diagonals of power series. Such representations are obtained by lowering the multiplicity of integration for the previously known integral representation. The procedure for reducing the order of integration is carried out in the framework of the Leray theory of multidimensional residues. The concept of the amoeba of a complex analytic hypersurface plays a special role in the construction of new integral representations. Keywords: multidimensional power series, complex integral, integral representation, amoeba, Taylor series, diagonal of a power series.

Citation: A.V. Senashov, A List of Integral Representations for Diagonals of Power Series of Rational Functions, J. Sib. Fed. Univ. Math. Phys., 2021, 14(5), 624-631. DOI: 10.17516/1997-1397-2021-14-5-624-631.

Introduction

A range of problems associated with branching of parametric integrals is concerned with a study of the diagonals of power series [1,2] and [3]. It should be noted that much earlier the concept of the diagonal of a power series was used by A. Poincare [4] to study the anomalies of planetary motion.

The diagonal of a Laurent power series

F(z) = £ caz« (1)

is defined as the generating function of a subsequence of coefficients [ca]aeL numbered by elements a of some sublattice L c Zn (see [1] and [5]). Such diagonals are called complete. Diagonals are graded according to the dimension (rank) of the sublattice.

Following [1], we describe the specifics of the problem on the properties of the diagonals of series for rational functions of n variables

F (z) = P(zl= P (zl,---,zn) , (2)

Q(z) Q(z1,...,zn)'

where P and Q are irreducible polynomials. Consider an arbitrary Laurent series for F centered at zero:

F (z) — caz "У \ ca1,...,an Z\1 ■■■za"

aezn aezn

* asenashov@mail.ru © Siberian Federal University. All rights reserved

It is known that such a series converges in domain Log-1(E), where E is a connected component of the complement Rn\Aq of amoeba of the denominator Q [6]. Recall that amoeba Aq of the polynomial Q or of the algebraic hypersurface

V = {z e (C\0)n : Q(z) = 0}

is called the image of V under the mapping Log : (C\0)n ^ Rn, defined by the formula

Log : (zi,...,z„) ^ (log |zi|,..., log \zn\).

Sometimes instead of the designation Aq we write AV. According to the result of the article [7], there is an injective order function

v : E ^ ZnQ Nq,

mapping each connected component E of the complement R"\Aq to integer vector v = v(E), belonging to the Newton polytope Nq of the polynomial Q. Thus, all connected components can be indexed as {Ev}, where v runs over a some subset of integer points from Nq. For example,

P

the Taylor series of a rational function —, Q(0) =0 converges in the component E0.

Q

Let us consider in more detail the p-dimensional diagonal of the series (1). Consider a p-dimensional sublattice l C L, with a basis q(1),...,q(p). We assume that this basis can be extended of L by n — p integer vectors q(p+1),..., q(n) (this assumption equivalent to say that the totality of all (p x p)-minors of the matrix A = (q(1),..., q(p)) are mutually prime) (see [8] or [9, Proposition 4.2.13]). Obviously the matrix

A = (q(1),...,q(n))

is unimodular, and we can assume it's determinant equals 1. Directions q(1),..., q(p) define a diagonals subsequence {clq}leZ+, where l ■ q means the product of the (1 x p)-matrix l and (p x n)-matrix A: Iq = l1q(1) + ■ ■ ■ + lpq(p). The generating function

dq (t)=Ys Clq t1 ■■■ tp

of the subsequence {clq}lez+ is called the one-sided q-diagonal of the series (1).

We assume that the denominator Q in (2) is not zero at z = 0, so the origin O e Zn belongs to the Newton polytope and there is nonempty component E0 of Rn \ Aq . We start by the Laurent series for the function (1) in Log-1(E0), which is in fact the Taylor series of (1) at z = 0. It is not hard to prove the following. If p e Log-1(E0), then dq(t) admits the integral representation

1 r zq(1) ■ ■ ■ zq(p) dzi dz

dq (t) = ^-V/ F (z)-^-^-...dzn, (3)

qW (2ni)nJrp (zq(1) —11) ... (zq(p) — tp) z1 zn'

where zq is a monomial zf1 ... zn, and cycle

r„ = {z e Cn : |z1| = ePl,..., |zn| = epn}

is chosen so that

a) poles of F(z) don't intersect the closed polydisc

Up = {z e Cn : |z1| < ePl,..., |zn| < epn};

b) parameters t = (t1,. ..,tp) satisfy the inequalities \ti \ < elqi 'p>, i = 1,. ..,p. Integration loop rp is a preimage Log-1 p of the point p from the connected component E0 of the amoeba Aq complements. Here we prove that the integral which represent the diagonal dq(t) admits a decrease of the order of integration while preserving the rationality of the integrand.

We will assume that Nq c Rn, and the image A-1(Nq) c Rn, here Rn and Rn are the n-dimensional real variable spaces u h v respectively. Let us denote by N' projection of the polyhedron A-1Nq on the coordinate (n — p)-dimensional plane {v G Rn : v1 = 0,... ,vp = 0}, and by Q'(t, w') Laurent polynomial Q[(t, w')A ] from variables w' = (wp+1,..., wn), wherein t1 , . . . , tp are parameters.

Theorem 1. Diagonal dq (t) in (3) is represented by an integral in the (n — p)-dimensional complex algebraic torus (C\0)n-p of variables wp+1,..., wn according to the formula

Log-1(p')

where

p' = ((Ap)p+i,..., (Ap)n)

belongs to the connected component E0 of the amoeba Aq' supplement of hypersurface V' = {w' G (C\0)n-p : Q'(t,w')=0}.

Proof of the theorem

Under the conditions of the theorem it is assumed that the diagonal (3) is considered for the

P

Taylor series of the rational function F = q . Therefore, it is automatically assumed that means

Q(0) = 0, and that means that the origin 0 is a vertex of the Newton polytope.

Because the determinant of the integer matrix A is equal to one, inverse matrix to it

A-1 =

W ...

\b1i] ... b^j

= j

(i) (i) is an integer and its elements by are algebraic complements to the elements qj . The rows and

columns of this matrix will be denoted by b(i) and bj respectively. Let us make in the integral

(3) the change of variables

(wbl,. .., w n ),

or, in a more detail:

b(1) b1n) b(1) b(n)

(zi, . . . ,zn) = (w11 ■■■ Wn1 ,w12 ■■■ Wn2 ,...,w

b(D b(n). 1 n ■■■ Wnn ).

First, note that zq will pass to Wi

q(i) q((i) q(i) ( b(() zq = z1( ... zn = [w,1 . . .W,

( b{() b(n) \q(i) (

w11 . . . wn1 . . .

b(1) b( ) q y1 n ... Wnn I

A

z = w

since {b(l), q(j)) = Sij is the Kronecker symbol.

Applying our change of variables to the logarithmic differentials, we obtain

j M b(1 b(n\ Ak) bi1 b(k)-1 b(n\,

dzi = d(w1i ...wn ) = 2^ k=1 bi w1i ...wki ...wn dwk

= b1 bF^ = b1 b^ .

w1i ... wn w1i ... wn

dzi

Multiplying the obtained expressions for the logarithmic differentials —- (taking into account

zi

the properties of the external product of differentials: dwidwi =0 h dwidwj = —dwjdwi), we get

i y^n b(^ — 1 y^n b(n)-1 |A |w1 i-1 ® ... wn i-1 i dw1 .. . dwn dw1 A ■ ■ ■ A dwn

E?-1 b(1) E?-1 bin) w1 ...wn

wi i-1 z ...wn z 1

Let us apply the formula for change of variables to the integral (3):

1 f A-1 w1 ■■■ wp dw1 ...dwn f

dq(t) = f0 F [(w1 ,...,wn) h---r--TT-, (5)

(2ni)n J (w1 — t1) ... (wp — tp) w1 ...wn

VtVp)

where ^ is the homomorphism induced by the mapping ^ : z ^ w = zA. The cycle rp is parameterized in the form

Log-1(p) = {z = ep+iA-1e : 6 e A([0, 2n)n)}.

Hence,

n(rp) = {w = zA : z e rp} = {w = eAp+AiA-d} = Log-1(Ap).

In this way,

Vi(rp) = {w : |w1| = e(Ap)1|wn| = e(Ap)n},

where (Ap)i is the i-th component of the vector Ap. By the Cauchy formula

1 f A-1 dw1 dwp dwp+1 ...dwn

dq(t) = f0 F[(w1,...,wn) ]-----------

(2ni)n J w1 —11 wp — tp wp+1 ...wn

Vt(Tp) we get

dq (t) = 17rn-p f F [(t1,...,tp,wp+1,...,wn)A-1 ] dWp+1 ■■■dwn , (6)

(2ni)n p J wp+1 . ..wn

Log-1 (p')

where

p = ((Ap)p+1,..., (Ap)n)

belongs to the connected component E0 of the complement of the amoeba Aq' of the hypersurface V' = {w' e (C\0)n-p : Q'(t,w') = 0}. The theorem is proved.

Let me make the following comment on the reduction of the formula (3) to (6). It is not difficult to see that the integrand in (3) admits representation in the form

J A ■ ■ ■ A f A ^

fi Jp

where ^ = ^p is a rational differential form of degree n — p, and fi = zq(i — ti. The system of binomial equations fi = 0,..., fp =0 defines an (n — p)-dimensional complex torus T"-p (embedded in the torus T" = (C \ 0)"). In this case, the real torus rp is a p-fold tube over a real torus y C T"-p(in the coordinates w, it is Log-1(Ap')). Thus, we are in the conditions of the multiple Leray residue formula (see [10,11]), according to that the integrals in (3) and (6) coincide.

Example

Consider the example of applying of the theorem to find the integral representation of the

H H

diagonal defined by the vectors qi = 111 and q2 = 2 of the Taylor series of the function

w w

F(z) =-. Newton polytope of the denominator of the function F has the

1 + Zi + Z2 + Z3 + Z2Z3 form shown in Fig. 1.

Fig. 1. Newton polytope 1 + zi + z2 + z3 + z2z3

For the two-dimensional diagonal dqi,q2 (ti,t2) = J2 ci1q(1)+i2q(2)tit2 in the set Log (E0),

one has the following integral representation

dq (t1,t2)

2 2

Z1Z2Z3 * Z1Z2Z3

dzi dz2 dzs

(2ni)3 J 1 + z\ + Z2 + Z3 + Z2Z3 (Z1Z2Z3 - ti)(ziz|z| - t2) zi Z2 Z3

rc

where the cycle

1

1

rp = {z e Cn : |z1| = eP1, M = ep2, |zs| = ep3}

is chosen so that

a) poles of F(z) don't intersect the closed polydisc

Up = {z e Cn : |z1| < ep1, ^ < ep2, |zs| < ep3};

b) parameters t = (t1,t2) satisfy the inequalities ^^ < elqi'p>, i = 1, 2.

/1 1 0\ i 2 —1 0\

Now let's form the matrix A = 11 2 0 1, then A-1 = 1—1 1 0 1. Using replacement

1 \1 2 V \0 —1 1/

zA = w, get z1 = w^w-1; z2 = w-1 wl^w-1; z3 = w3. The denominator of the function F after replacement is converted to 1 + wlw-1 + w-1w12w-1 + w3 + w-1w12 and Newtonian polytope is shown in Fig. 2

Fig. 2. Newton polytope 1 + w2w- 1 + w- 1w^w3i 1 + w3 + w- lw\ The integral after replacement zA 1 = w looks like this

w i * w2 dwi dw2 dw3

dq (ti,t2)

1

1

(2ni)3 J 1+ wjw-1 + w-1w1w-1 + w3 + w-1w1 (wi - ti)(w2 - t2) wi w2 w3

r o

After integration by the Cauchy formula with the variable w1 we obtain the following form of the diagonal

dq (ti,t2)

1

1

w2 dw2 dw3

(2ni)2 J 1+ t2w-1 + t- 1w1w- 1 + w3 + t-1wl w2 - t2 w2 w3

We construct the Newton polytope of the denominator of the function F(t1 ,w2, w3) (Fig. 3). After integrating with the variable w2 we obtain

r

Fig. 3. Newton polytope 1 + t2w.2 1 + t. iw^w3i 1 + w3 +t. iwi 1 i 1 dw^

dq(ti,t2) = J2ni) J ■ W■ (8)

Therefore the integral (7) admits a reduction to the one-dimensional integral (8) with rational integrand. It is known ( [12], Section 10.2) that such integral is an algebraic function in variables ti,t2.

This work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (no. 18-1-7-60-3).

References

[1] D.Pochekutov, Diagonal sequences of Laurent coefficients of meromorphic functions of several variables and their application, Dis. Cand. phys-mat. nauk: 01.01.01, Krasnoyarsk, 2010 (in Russian).

[2] J.Denef, L.Lipshitz, Algebraic power series and diagonals, J. Number Theory, 26(1987), 46-67.

[3] L.Lipshitz, D-finite power series, J. of Algebra, 122(1989), 353-373.

[4] A.Poincare, Selected Works in Three Volumes. Vol. I. New Methods of Celestial Mechanics, Moscow, Science, 1971 (in Russian).

[5] D.Pochekutov, Diagonals of Laurent series of rational functions, Sib. mat. zhurn., 50(2009), no. 6, 1370-1383 (in Russian).

[6] I.Gelfand, M.Kapranov, A.Zelevinsky, Discriminants, Resultants and Multidimentional Determinates, Boston, Bikhauser, 1994.

[7] M.Forsberg, M.Passare, A.Tsik, Laurent determinants and arrangements of hyperplane amoebas, Advances in mathematics, 151(2000), no. 1, 45-70.

[8] L.Nilsson, M.Passare, A.Tsikh, Domains of Convergence for A-hypergeometric Series and Integrals, Journal of Siberian Federal University. Mathematics & Physics, 12(2019), no. 4, 509-529. DOI: 10.17516/1997-1397-2019-12-4-509-529.

[9] T.Sadykov, A.Tsikh, Hypergeometric and algebraic functions of many variables, Moscow, Nauka, 2014 (in Russian).

[10] A.Tsikh, A.Yger, Residue currents, J. Math. Sci. (N.Y.), 120(2004), no. 6, 1916-2001.

[11] L.Aizenberg, A.Yuzhakov, Integral representations and deductions in multidimensional complex analysis, Novosibirsk, Nauka. Sibirskoye Otdeleniye, 1979 (in Russian).

[12] A.Tsikh, Multidimensional residues and their applications, Novosibirsk, Nauka, 1988 (in Russian).

Список интегральных представлений для диагонали степенного ряда рациональной функции

Артем В. Сенашов

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В работе приводятся интегральные представления для диагоналей степенных рядов. Такие представления получаются понижением кратности интегрирования для известного ранее интегрального представления. Процедура понижения кратности реализуется в рамках многомерной теории вычетов Лере. Особую роль в конструкции новых интегральных представлений играет понятие амебы комплексной аналитической гиперповерхности.

Ключевые слова: многомерные степенные ряды, комлексный интеграл, интегральное представление, амеба, ряд Тейлора, диагональ степенного ряда.