ANALYTIC CONTINUATION OF DIAGONALS OF LAURENT SERIESFOR RATIONAL FUNCTIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-360-368 УДК 517.55

Analytic Continuation of Diagonals of Laurent Series for Rational Functions

Dmitry Yu. Pochekutov*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 11.12.2020, received in revised form 01.01.2021, accepted 25.03.2021 Abstract. We describe branch points of complete q-diagonals of Laurent series for rational functions in several complex variables in terms of the logarithmic Gauss mapping. The sufficient condition of non-algebraicity of such a diagonal is proven.

Keywords: diagonals of Laurent series, hyperplane amoeba, logarithmic Gauss mapping, zero pinch, monodromy.

Citation: D. Yu. Pochekutov, Analytic Continuation of Diagonals of Laurent Series for Rational Functions, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 360-368. DOI: 10.17516/1997-1397-2021-14-3-360-368.

1. Introduction and preliminaries

We use the notation Cx for the one-dimensional complex torus C \ {0}. For vectors z = = (z1,..., zn) in Cn or (Cx )n and a = (ai,... ,an) in Zn, denote by za the monomial z^1 ... zf™.

P (z)

Consider a Laurent series for a rational function F(z) = of n complex variables centered

Q(z)

at the origin:

F(z) =J2 C*za. (1)

Let L C Zn be a sublattice of the n-dimensional integer lattice. Then the generating function for the subsequence {Ca}aeL of the coefficients indexed by L is called the complete diagonal of the Laurent series (1). Throughout the paper, we consider the sublattice of rank 1 generated by the irreducible vector q = (q1,... ,qn) from Zn \ {0}. We will call the corresponding diagonal

dq (t)= Y, Cq ktk

k=-œ

a complete q-diagonal of the Laurent series (1). Such a diagonal can be written naturally as a sum of two subseries d+ (t) and d- (t) with only non-negative and negative powers of t, correspondingly. We call them one-sided q-diagonals. Clearly, we have the equality dq(t) = d+(t) in the case of Taylor series. For the unit vector I = (1,..., 1), we denote di(t) by d(t), and refer to /-diagonal simply

as a

diagonal.

Further, we consider irreducible polynomials P(z) and Q(z). It is well-known that domains of absolute convergence of power series are logarithmically convex. In the case of the Laurent

* dpotchekutov@sfu-kras.ru https://orcid.org/0000-0002-4545-2129 © Siberian Federal University. All rights reserved

series (1), it is convenient to use the notion of an amoeba of the denominator Q(z) of the rational function F(z) in the description of such domains. Recall [1, Section 6.1] that the amoeba of a polynomial Q is the image of a hypersurface ZX(Q) under the logarithmic mapping A : (Cx )n ^ Rn defined by

A(z) = (log |zi|,..., log \zn\),

where ZX(Q) is defined in the complex torus (Cx)n by zeroes of the polynomial Q.

The complement Rn\Aq consists of a finite number of connected components E that are open and convex. The preimages A-1(E) of these components are domains of absolute convergence for Laurent expansions (1) (centered at the origin) for the rational function F(z) (see Section 2). Amoebas are closely related to the notion of the logarithmic Gauss mapping

Yq : regZx(Q) ^ CPn-1

defined as

YQ<z)= (zi g<zZn g w) (2)

in regular points z of the hypersurface Zx (Q). In fact, the set of critical points of the logarithmic projection A : Zx(Q) ^ Rn contains the boundary OAq and coincides with Y-1(RPn- ).

The complete q-diagonal dq(t) of the Laurent series (1) that converges in the domain A-1(E) for a rational function F can be represented as the integral (see Section 2)

1 f P(z) zq dz1 A ... A dzn

dq (t)

(2ni)n J r Q(z) zq - t z\ ...zn

over the n-dimensional cycle r = A-1(y2) — A-1(y1) in (Cx)n \ {Zx(Q ■ (zq — t))}. The parameter t in the integral representation is chosen so that the amoeba of the polynomial zq - t (that is the hyperplane (q, u) = log |t| with the normal vector q) divides the component E into two parts, and points y1, y2 are chosen from different parts of this partition. The ramification of the complete q-diagonal happens when a value of the parameter t is such that the rank of the n-dimensional homology group (Cx)n \ {Zx(Q ■ (zq — t))} drops.

Since E is convex, the restriction of a linear function (q, u) to the closure of E in Rn attains extreme values on the boundary dE. Let u0 = u0(q) be one of the points of the boundary dE such that the function specified above attains an extreme value. Then the branch points of dq(t) should be among points of the form pq, where p = p(q) is a point of the hypersurface Zx(Q) such that A(p) = u0.

The main result of the present paper is the theorem that characterises branch points of diagonals.

Theorem 1. Let the Laurent series (1) for a rational function of n variables converge in the domain A-1(E), and let dq(t) be its complete q-diagonal. If q = Yq(p), where the point p is regular for the logarithmic Gauss mapping and A(p) € dE, then

1. In the case n = 2k the point t0 = pq is a branch point of finite order 2 of dq (t).

2. In the case n = 22k + 1 the point t0 = pq is a branch point of infinite order (logarithmic branchpoint) of dq (t).

In the context of enumerative combinatorics (see. [2, Section 6.1]), there is the following hierarchy of generating functions

{rational} C {algebraic} C {D — finite}.

It was proven in [3] that complete q-diagonals of Laurent series for rational functions of two complex variables are algebraic. In expositions that deals with diagonals (see, for instance, [4, Section 2] or [2, Section 6.3]), treatment of the case of more than two variables is limited by pointing at the example of non-algebraic diagonal of the Taylor series for the rational function of three variables.

Since algebraic functions cannot have branch points of infinite order, Theorem 1 gives the sufficient condition of non-algebraicity of a diagonal in the case when the dimension n is odd.

Corollary 1. Let the Laurent series (1) for a rational function of 2k + 1 variables converge in the domain A-1 (E), and let dq(t) be its complete q-diagonal. If q = yq (p), where the point p is regular for the logarithmic Gauss mapping and A(p) € dE, then dq(t) is a non-algebraic function.

2. Amoebas and integral representation for diagonal

From the moment of diagonals appeared on the mathematical scene (see [5, p. 280]), the important role in their study was played by integral representations. George Polya showed the algebraicity of a diagonal of a bivariate rational Taylor series from a particular class in [6]. His proof was based on a representation of the diagonal by an integral over a contour in the complex plane. Exploiting a similar idea it was shown in [4, 7] that the diagonal of an analytic power series F in a bidisk {\z1\ < A, \z2\ < B} can be represented as

i(t) = hL

z i\ dZ

where e = ^A + B^J/2. If, in addition, F converges to a rational function, then evaluating the integral by residues gives that the diagonal is algebraic, see [4, Section 2] and [2, Section 6.3]. Further, in [8] it was proved that the q-diagonal of the Taylor series for a rational function P (z)

F(z) = , of n complex variables holomorphic at the origin has the integral representation Q(z)

1 f P (z) zq-J ,

dq (t) = 7-— —-dz,

qK> (2ni)n J-pp Q(z) zq - t '

where the cycle = {z € Cn : \z1 \ = p1,..., \zn\ = pn} is chosen so that the closed polydisk {\z1 \ < p1,..., \zn\ < pn} contains no poles of the function F(z), and pq > \t\. It will be convenient for us to use the following notation

1 P(z)zq-Jdz.

(2ni)

In order to describe the integral representation for a complete q-diagonal of the Laurent series (1), we list necessary facts about amoebas of polynomials.

Recall that the Newton polytope Aq of a polynomial Q is the convex hull in Rn of the set of exponents of the monomials occuring with non-zero coefficients in Q. According to Propositions 2.4-2.6 in [9], on the set {E} of connected components of Rn \ Aq there exists an injective order mapping

v : {E}^ Aq n Zn

such that the dual cone to Aq at the point v(E) coincides with the recession cone of the component E. Then it follows from this fact that the number of connected components of the amoeba complement is at most equal to the number of integer points in Aq (see [9, Theorem 2.8]). Note that the proof of the injectivity of v also establishes that components E are convex in Rn. Corollary 1.6 in [1] says that all centered at the origin Laurent expansions (1) of a rational P(z)

function F(z) = q ) are in a bijective correspondence with the connected components {E}.

The sets A-1(E) are the convergence domains for the corresponding Laurent expansions. If the rational function F(z) is holomorphic at the origin, then its Taylor expansion converges in the domain A-1(E), where v(E) = (0,..., 0), and the point (0,..., 0) is a vertice of the Newton polytope A q .

The following proposition from [3] generalizes the integral representation for diagonals of Taylor series that have been mentioned above.

Proposition 1. Let the Laurent series (1) for a rational function of n variables converge in the domain A-1(E), where E is a connected component of the complement Rn \ Aq, and let y1; y2 are points in E such that the inequality (q, y1) < (q, y2) holds for a non-zero q € Zn. Then the complete q-diagonal dq(t) of the Laurent series (1) has the integral representation

® = i QzUq—7> • <3>

where (q, y^ < log t < (q, y2), and r = A-1(y2) — A-1(y1).

3. Proof of Theorem 1

Note that the differential form w is regular in (Cx)n, while the differential form in the integral representation (3) is meromorphic in (Cx)n with polar singularities on hypersurfaces

S1 = Z x(Q), S2 = Z x(zq — t).

Let y1, y2 be points in E chosen as specified in Proposition 1. The fibers A-1(y1), A-1(y2) of the logarithmic projection over these points are n-dimensional real tori (Cx)n that define classes in the reduced homology group Hn((Cx)n \ S1 U S2) with compact supports.

We want to show that the family {S1,S2} has a so-called quadratic zero-pinch (see [10, Section IV.1]) at the point p for t = t0, where t0 = pq. For this purpose, we introduce new coordinates w = (w1,... ,wn) in the n-dimensional torus (Cx)n.

We first note that since vector q is irreducible, according to the Invariant Factor Theorem (see [11, Theorem 16.6]), there exists an unimodular transformation A : Zn ^ Zn that takes vector q to vector e1 = (1, 0,..., 0). This transformation induces the diffeomorphism (Cx)n ^ (Cx)n defined as

w1 = zai,... ,wn = zan,

where aj's are columns of the matrix for the transformation A, and a1 = q. In new coordinates, the hypersurfaces S1, S2 are defined by equations

Q(w) =0, w1 — t = 0,

correspondingly.

Next, assume, without loss of generality, that QW1 (p) = 0, where the point p = (pai,..., pan). Then, by the Implicit Function Theorem, there exists a sufficiently small neighbourhood U of the point p such that S^ is defined in U as a graph of some analytic function,

Si П U = {w e U : wi = f (w2,... ,wn)}.

Therefore, the intersection Si П S2 is defined in U as the zero set of the system

( wi - f (w2,..., wn) =0, \ wi - t = 0.

From the definition of the logarithmic Gauss mapping (2), it follows that

Y q (w) = (1 : -w2fw2 (w2,.. .,wn) : ... : -wnfwn (w2,.. .,wn))

for w e U. In particular, Yq (p) = (1:0... : 0). Since the (i, j)-component of the Jacobian matrix of the logarithmic Gauss mapping Yq at the point p e U has the form

(-pifWiWj (p2 , . . . ,pn ^ , i,j =2,...,n,

the Jacobian determinant of Yq at p and the Hessian determinant of the function f (w2,..., wn) at the point (p2,... ,pn) vanish simultaneously. If p is a regular point of yq then p is a regular point of Yq . So the point (p2,... ,pn) is a Morse critical point for the function f (w2,... ,wn), and by the Morse lemma, there exist local coordinates (w2,... ,wn) in a neighbourhood of this point such that f = w2 + ... + wn + pq. So the intersection Si П S2 is given locally by the equation

w2 +... + w2n + pq -1 = 0.

Therefore, the family of the hypersurfaces Si , S2 has the quadratic zero-pinch at the point p for t = pq.

Thus, for the discriminant value t0 = pq of the parameter t, we have the standart degeneration of type Pi = P2 (in terms of the notation of [12, Section I.8]). The monodromy operator

Ф : Hn((Cx)n \ Si U S2) ^ Hn((Cx)n \ S\ U S2),

defined by a small loop going around t0 was calculated in [10, Part IV]. This operator reduces to the standart Picard-Lefschetz formula for the Morse singularity in <C,n-i+i = Cn-i. So, by Theorem 2.4 in [10, Part IV], we have that

Ф([Г]) = [Г] + t[E]

where 1 is a non-zero integer, and the class [£] is defined as follows. According to the Thom Isotopy theorem, the monodromy acts identically outside a sufficiently small neighbourhood W of the point p. Let a be the vanishing sphere of the dimension n — 2 in the intersection of Si П S2 and W. Then [£] = i^S2[a], the homomorphism iФ is induced by the inclusion of W into (Cx)n, and S2 : Hn-2(Si П S2 П W) ^ Hn(W \ (Si U S2)) is 2-iterated coboundary operator of Leray defined in Theorem 2 of [10, Part II].

Note that the Picard-Lefschetz formula also gives us

Ф(Щ) = (-1)п-1[П.

Knowing the transformation of [r] and [E] by $ allows us to continue the integral

dq (t) = I Q(z)"zq -1) analytically along a small loop around the point t0. Let

q(t) f '

/E Q(z)(zq -t)'

Then during one traversal of the mentioned loop the integral for dq(t) goes to

dq (t) + iq(t).

If the dimension n = 2k, the two traversals of the loop give

dq (t) + iq(t) + (-1)2k-hq(t) = dq (t).

So, the point to is a branch point of order 2 for the diagonal dq(t). If the dimension n = 2k + 1, the two traversals of the loop give

dq (t) + iq(t) + (-1)2k iq(t) = dq (t) + 2iq(t).

In this case, t0 is a branch point of infinite order for the diagonal dq(t). The theorem is proved.

4. The diagonal of the multivariate geometric series

The purpose of this section is to illustrate Theorem 1.

Consider the polynomial L(z) = 1 — z1 — ... — zn. The multivariate geometric series

1 = E ("1 + .. + an)' ~a

aEN

converges in the domain A~1(E0), where E0 is the compoment of the complement Rn \ AL that corresponds to the constant term of L.

For convenience, we denote the diagonal of this Taylor series by

dn(t) = E nn tk. (4)

k=0 V ''

The logarithmic Gauss mapping yl : Zx (L) ^ CPn-1 is a birational isomorphism with the inverse given by

qj ■ 1

zj = -:-:-, j = 1,...,n

q1 + ... + qn

where q = (q1 : ... : qn) € CPn-1. Also, the point p = (—,..., —) is projected by the

nn

logarithmic mapping A to the point of the boundary dE0, so that, by Theorem 1, the point t0 = p1 = 1/nn is a branch point for the diagonal dn(t), and the type of this branch point depends on the parity of n. We note that

1

d2(t)

VT—4t'

by means of the generalized binomial expansion. Thus, the diagonal d2(t) is an algebraic function that has a branch point of the order 2 at t0 = 4.

In the case n = 3, the diagonal (4) is represented by the Gaussian hypergeometric function

ds(t) = 2F1 (1, 3, 1;27t) ,

so that t0 = 27 is a branch point for the diagonal. Note that the parameters of this hypergeometric function are not in Schwarz's list of the cases when the Gaussian hypergeometric function is algebraic.

Proposition 1. The diagonal d3(t) has the form

ds(t) = as(t)log(1 - 27t) + b3(t),

in a neighbourhood of the point t0 = —, where the functions a3(t) and b3(t) are holomorphic and non-vanishing at the point t0 = 27.

Proof. According to [13, Section 16], we can write the hypergeometric function 2 F1(1, §, 1;27t)

ng to [13, Section 16], we can write the hypergeometric function 2^1 , 2

as the integral

2 1 2

(Ijrfö J r2(-z)r(3 + z)r(I+c)(i -mzdz

2™ r(3)rV3,

- §+iR

with the meromorphic integrand that has three groups of poles

12

Zk = k, Zk = - 3 - k, nk = - 3 - k, k € N U{0}.

The poles £k lie on the complex plane to the right of the integration contour, while the poles Zk, nk lie to the left of it.

Evaluating the integral as the sum of residues in poles £k of the first group gives us the desired representation. □

Further, it is clear from the representation

d4(t) = 3F2 (4, §, 3;1,1;256t)

in the form of the generalized hypergeometric function that the diagonal d4(t) has a branch point 1

at t0 =-.

0 256

By a happy coincidence, the generalized hypergeometric function 3F2 that corresponds to this specific set of parameters can be written in the form

d4(t) = (F (1, §;1;256t))2 (5)

with a help of Clausen's formula [14]. It allows us to describe a type of the branch point t0 = -1-

256

in a way that is similar to the proof of Proposition 1. Proposition 2. The diagonal d4(t) has the form

d4(t) = a4(t)(1 - 256t)1 + b4(t),

in a neighbourhood of the point t0 = 256, where functions a4(t) and b4(t) are holomorphic and

non-vanishing at the point t0 = ——.

256

Proof. According to [13, Section 16], we can write the hypergeometric function 2 F1(1, §, 1;256t) as the integral

r(8)r(7) 1

J r(-c)r(2 -z)r(8 + z)r(8 + z)(i - 256t)zdz

— Hq t ^

where the integration contour separates poles of the function r(— Z\ — () of the form

6 = k, Zk = 8 + k,

from the poles of r( 1 + Z) r( § + Z) of the form

1 k 3 k

nk = — 8 — k, Kk = — ^ — k.

The parameter k ranges over the set N U {0}.

We let b(t) denote the sum of residues of the integrand at the point . It occurs that b(t)

is holomorphic at t0 = and does not vanish at this point. At the same time, the sum of

residues of the integrand at the points Zk has the form a(t)(1 — 256t)1/2, where the function a(t)

is holomorphic at t0 = —— and is non-vanishing at this point. 256

Thus, the function 2F1 (g, §, 1;256t) has the representation

a(t)(1 — 256t)1 + b(t)

in some neighbourhood of the point t0 = -. Then the Proposition follows directly from the

256

Clausen formula (5). □

The research is supported by grant of the Russian Science Foundation (project no. 20-1120117).

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Аналитическое продолжение диагоналей рядов Лорана рациональных функций

Дмитрий Ю. Почекутов

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Мы описываем точки ветвления полных д-диагоналей рядов Лорана рациональных функций нескольких комплексных переменных в терминах логарифмического отображения Гаусса. Доказано достаточное условие неалгебраичности такой диагонали.

Ключевые слова: диагонали рядов Лорана, логарифмическое отображение Гаусса, амеба гиперповерхности, нулевой пинч, монодромия.