Analytic continuation for solutions to the system of trinomial algebraic equations Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2020-13-1-114-130 УДК 517.55+512.761

Analytic Continuation for Solutions to the System of Trinomial Algebraic Equations

Irina A. Antipova* Ekaterina A. Kleshkova^ Vladimir R. Kulikov*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 05.10.2019, received in revised form 09.11.2019, accepted 24.12.2019 Abstract. In the paper, we deal with the problem of getting analytic continuations for the monomial function associated with a solution to the reduced trinomial algebraic system. In particular, we develop the idea of applying the Mellin-Barnes integral representation of the monomial function for solving the extension problem and demonstrate how to achieve the same result following the fact that the solution to the universal trinomial system is polyhomogeneous. As a main result, we construct Puiseux expansions (centred at the origin) representing analytic continuations of the monomial function. Keywords: algebraic equation, analytic continuation, Puiseux series, discriminant locus, Mellin-Barnes integral.

Citation: I.A.Antipova, E.A. Kleshkova, V.R.Kulikov, Analytic Continuation for Solutions to the System of Trinomial Algebraic Equations, J. Sib. Fed. Univ. Math. Phys., 2020, 13(1), 114-130. DOI: 10.17516/1997-1397-2020-13-1-114-130.

Introduction

We consider a system of n trinomial algebraic equations of the form

£ af£ya = 0,i = l,...,n, (1)

with the unknown y = (yl,...,yn ) G (C \ 0)n and variable coefficients a(i), where c Zn are fixed three-element subsets and ya = y■... ■ is a monomial. Without loss of generality we assume, that all sets A(i contain the zero element 0 (this may be achieved by dividing the ith equation in (1) by a monomial with the exponent in

A« see the system (2) below). We call (1) the universal trinomial system since any trinomial algebraic system is a result of the substitution of polynomials in new variables for coefficients a.

When n =1, the system (1) is a scalar trinomial equation. It has a special place in the centuries-old history of algebraic equations. As early as 1786, Bring proved that every quintic polynomial could be reduced to the trinomial form y5 + ay + b using the Tschirnhaus transformation. At the turn of the XIX-XX centuries, the dependence of norms of roots on coefficients

* [email protected] https://orcid.org/0000-0003-1382-0799 [email protected] https://orcid.org/0000-0002-7443-2979 [email protected] https://orcid.org/0000-0002-2291-0449 © Siberian Federal University. All rights reserved

of the trinomial equation with fixed support was actively studied. Although algebraic characterisation of the mentioned dependence was given by Bohl already in 1908, the geometric view on the problem has been formed much later. In the recent study by Theobald and de Wolff [15], a geometrical and topological characterisation for the space of univariate trinomials was provided by reinterpreting the problem in terms of the amoeba theory.

Of particular interest is the reduced system of n trinomial equations

y^ + x^ - 1 = 0,« = 1,...,n, (2)

with the unknown y = (y1,..., yn), equation supports A(i) := {w(i),a(i), 0} c Z" and variable complex coefficients x = (x1,... ,xn). It is assumed that a matrix w composed of column vectors w(1),..., w(n) is nondegenerate.

Let y(x) = (y1(x),... ,yn(x)) be a multivalued algebraic vector-function of solutions to the system (2). We call a branch of y(x) defined by conditions yi(0) = 1, i = 1,... ,n the principal solution to the system (2). Having determined the principal solution y(x), we consider the following monomial function

yd(x):= yd (x) ■ ... ■ ydnn (x), d =(di ,...,dn) € Z+. (3)

Our goal is to obtain Puiseux expansions (centred at the origin) representing analytic continuations of the Taylor series for the monomial yd(x) of the principal solution to the system (2). Puiseux type parameterizations of an algebraic variety via the amoeba of the discriminant locus of the variety canonical projection were studied in [6]. The existence of such parameterizations for plane curves was proved by Puiseux [12]: this fact is known as the Newton-Puiseux theorem which states that one can find local parameterizations of the form x = tk, y = f(t), where f is a convergent power series. We aim at investigating Puiseux expansions for analytic continuations of (3) which may fail to "recognize" some pieces of the discriminant set. It means that the convergence domain G of a series projects onto the domain Log(G) containing a certain collection of connected components of the discriminant amoeba complement. An example in Section 1 illustrates how the series converging in the preimage Log-1(E0) of the component E0 of the amoeba complement admits an analytic continuation to the domain G for which Log(G) covers components E1, E2 and an amoeba tentacle separating them, see Fig. 1. This analytic continuation is given by another series expansion.

When n = 1, analytic continuations for the Taylor series of the principal solution to the universal algebraic equation (not necessarily a trinomial) were found in [3], where the Mellin-Barnes integral representation for the solution was used as a tool of the analytic continuation. This integral, with indicating the convergence region of it, was wholly studied in [2]. While a power series converges in a polycircular domain, a Mellin-Barnes integral converges in a sectorial domain which is defined only by conditions for arguments arg xi of variables xi. Remark that the intersection of these domains is always nonempty. Consequently, a series expansion of the solution to the equation admits an analytic continuation into the sectorial domain by means of the integral. Of course, we may follow this approach to getting analytic extensions for the monomial (3) in a case when the corresponding Mellin-Barnes integral represents it. Herein we can obtain analytic continuations of the Taylor series in the form of Puiseux series via the multidimensional residues technique.

However, we can get the same series following the fact that the solution y(a) to the system (1) is polyhomogeneous. This means that via some monomial transformation of coefficients the system (1) can be reduced to the form (2) or to another system which, similarly, has only one

variable coefficient in each equation. We perceive any reduced system of equations as the general (homogeneous) system (1) written in suitable coordinates. The transition from one reduced system to another enables us to obtain series continuations for monomials of coordinates of solutions to these systems.

The paper is organized as follows. In Section 1 we review the technique of the calculation of multidimensional Mellin-Barnes integrals which is based on the separating cycle principle formulated in [16] (see also [17]). We present an example which illustrates what computational issues can arise in this way of getting analytic extensions. In Section 2 we discuss the procedure of the dehomogenization (reduction) of the system (1) and obtain the Taylor series expansions for the monomials of the principal solutions to all reduced systems associated with the system (1). Theorem 1 gives these series as a result of the application of the logarithmic residue formula [5] and the linearization procedure for each reduced system. The idea of using the logarithmic residue formula for getting the Taylor expansions was developed in [8], where the special instance of the reduced polynomial system with the diagonal matrix w was considered. In Section 3 we use Taylor expansions derived in Theorem 1 and appropriate monomial transformations to obtain the desired Puiseux series which are supposed to be the analytic continuations of the Taylor series for the monomial yd(x) of the principal solution to (2) (Theorem 2). Finally, we discuss the example from Section 1 again in terms of the result of Theorem 2.

1. Mellin-Barnes integral as a tool of analytic continuation

Traditionally, the Mellin-Barnes integrals are regarded as the inverse Mellin transform for special meromorphic functions, which are rations of products of a finite number of superpositions of gamma functions with affine functions. Their role in the theory of algebraic equations was revealed first by Mellin in [9], where he wrote down without any proof the integral representation for the solution to the universal algebraic equation later investigated in [2]. In our study we consider such integrals in the extended sense, having in mind the presence of a polynomial factor in the integrand besides gamma-functions.

The Mellin integral transform for monomials of a solution to the reduced polynomial system was studied in [1] and [14]. Following [14], we associate the Mellin-Barnes integral

l r n r(zj)r(j - -1 (a- ,z))

1 H ( I 1 " " j \ Q(z)x-Z dz (4)

(2ni)

" i-i r(^ - -1 (aa,z) + zA +

Y+iRn j-i y^j Uj^ I j J

with the monomial yd(x). In (4) x z denotes the product x-Zl • ... • xnZn, Uj is the jth row of the matrix u composed of column vectors belongs to the domain

U = {u e R+ : (aj ,u) <dj, j = 1,..., n},

and Q(z) is a polynomial represented by the determinant

Q(z) = -r^- det Sj(dj - (a-,z)) + a(i)zi det w

ij-i

where Sj is the Kronecker symbol. Here it is assumed that w is a diagonal matrix with elements w1,... ,wn on the diagonal.

Remark that the integral (4) can have the empty convergence domain. It follows from [7] that its convergence domain is nonempty if and only if all the diagonal minors of the matrix u

n

are positive. In this case, the integral (4) represents the monomial yd(x) of the principal solution to the system (2), and it can be used as a tool of the constructive analytic continuation of power series.

Let us show how to calculate the integral (4). A method of the calculation is based on the separating cycle principle formulated in [16] and developed in [17]. This principle deals with the calculation of the Grothendieck-type integrals

1 ¡' h(z)dz

(5)

(2ni)n J fi (z) ...fn(z)

Ag

where the integration set Ag is the skeleton of the polyhedron ng associated with the holomorphic proper mapping g : (g1,...,gn) : Cn ^ Cn, and the integrand has poles on divisors Dj = = {z : fj(z) = 0}, j = 1,...,n. The polyhedron ng is the preimage g-1(G) of the domain G = G1 x ... x Gn, where each Gj is a domain on the complex plane with the piecewise smooth boundary. We associate a facet aj = {z : gj (z) € dGj, gk(z) € Gk, k = j} of the polyhedron ng with j € {1,..., n}.

Definition. A polyhedron ng is said to be compatible with the set of divisors {Dj} if for each j = 1,... ,n the corresponding facet aj of the polyhedron ng does not intersect the divisor Dj .

Assume further that the intersection Z = D1 n ... n Dn is discrete. The local residue with respect to the family of divisors {Dj} at each point a € Z (the Grothendieck residue) is defined by the integral (see [16])

res'-Q = ^ / Q

r a(f )

where Q is the integrand in (5), and ra(f) is a cycle given in the neighborhood Ua of the point a as follows

ra(f) = {z € Ua : \h(z)\ = E1,..., \fn(z)\ = En}, £j << 1.

If a is a simple zero of the mapping f, i.e. the Jacobian Jf = df/dz is nonzero at the point a, then the local residue is calculated by the formula

h(a)

res f,aQ = . (6)

Jf(a)

Theorem 1 (principle of separating cycles). If the polyhedron ng is bounded and compatible with the family of polar divisors {Dj} then the integral (5) equals to the sum of Grothendieck residues in the domain ng.

One can reduce the integral (4) to the canonical form (5) in the following way. We interpret the vertical integration subspace 7 + iRn as the skeleton of some polyhedron. For instance, in the case n = 1, it can be the skeleton of only two polyhedra: the right and left halfplanes with the separating line 7 + iR. For n > 1 this subspace may serve as the skeleton of an infinite number of polyhedra. Our objective is to divide all the set of 2n families of polar hyperplanes of the integral (4)

Lj : zj = -v,

dj 1

Ln+j :---{aj,z) = -v, j = 1,...,n, v € Z"

wj wj

into n divisors and construct a polyhedron compatible with this family of divisors. We consider polyhedra of the type

ng = {z € Cn : Regj(z) < rj, j = 1,...,n},

where gj (z) are linear functions with real coefficients. It is clear that ng = n + iRn where n is a simplicial n-dimensional cone in the real subspace Rn C Cn. Remark that in the case of an unbounded polyhedron, besides the compatibility condition of the polyhedron and polar divisors, one should require a sufficiently rapid decrease of the integrand Q in the polyhedron ng. For the integral (4) the nonconfluence property provides the decrease of the integrand, see [10] and [17]. We recall that the nonconfluence property for the hypergeometric Mellin-Barnes integral means that sums of coefficients of the variable zj over all gamma-factors in the numerator and the denominator are equal.

Now, applying the technique discussed above, we construct analytic continuations for the solution to the following system of equations

j y4 + x1y2y2 -

I 4 , 2 1 O (7)

+ x2y1y2 - 1 = 0.

For the description of the convergence domains of power series and Mellin-Barnes integrals we introduce the following mappings from (C \ 0)n into Rn:

Log : (x1, ...,xn) —> (log \x1\,..., log \xn\),

Arg : (x1, ...,xn) —> (arg x1,..., arg xri).

The monomial y1(x) ■ y2(x) of the principle solution to the system (7) admits the Taylor series representation

E -Tr( 1 + 1 k1 + 1 kf(\ + 1k1 + 1 k2) 1(1 + k1 + ^#, (8)

¿0 k! r( 4 - 1 k1 + 1 k2) r( 4 + 1 k1 - 1 k2) ^ 1 2'

which converges in some neighborhood of the origin, see Theorem 1 below. In turn, the Mellin-Barnes integral of the form

1 r r(z1)r(z2)r(4 - 1 zi - 4Z2) r(4 - 4zi - 1 z2) (1 - zi - z^)

(2ni)2 J * r(4 + 1 z1 - 1 z2) r(4 - 1 z1 + 1 z2) 16

Y+iR2

where 7 is a point in the open quadrangle

U = {u € R+ :2u1 + u2 < 1, u1 + 2u2 < 1} , represents the monomial y1(x) ■ y2(x) in a sectorial domain Arg_1(©) determined by

(9)

Q=^(eue2) G R2 : |0i| <2, N<n, - 0i\ < ï, 02 - 2011 < ^ (10)

2' 121 2" 2 11 4 " 2 11 4

here 01 = arg x1,02 = arg x2. Fig. 2 shows the domain © which is the interior of the convex octagon. The general description of convergence domains of multiple Mellin-Barnes integrals gives Theorem 4.4.25 in the book [13]. Thus, the integral (9) gives the analytic continuation of the series (8) into the sectorial domain Arg_1(©) .

We next calculate the integral (9) using the principle of separating cycles. It admits a representation as a sum of local residues of the integrand

Q

r(*i)r(*2)r (4 - 1 Zi - 4Z2) r (4 - 4Z1 - 1 z2) (1 - zi - Z2)

r (4 + 2Zi - iZ2) r (5 - 4Zi + iZ2)

16

x-Zl x2Z2 dZ1 dZ2 (11)

in some polyhedron, which contains the vertical imagine integration subspace 7 + iR2 as the skeleton. Furthermore, the polyhedron and polar divisors of Q should satisfy the compatibility conditions.

The form Q has four families of polar complex lines:

Li : Zi = -v,

L2 : Z2 = -v,

L3 : 4 - 4(2z1 + Z2) =

L4 : 4 - j(Zi + 2Z2) = -v, v G Z>.

(12)

Figs. 3 and 4 show the intersection of the real subspace with families (12), and also with

r 5 1 1

L5 : J + ^- Jz2 =

5 1

1

L6 : ~A tZi + -Z2 = -v,

4 4

which are polar sets of gamma-functions in the denominator of the form (11). The quadrangle U, to which the point 7 belongs, is coloured in grey.

Fig. 1. The discriminant amoeba of the system (7) and its complement components Eu

Fig. 2. The domain 8

First, given the set of all polar lines of the integrand Q, we form two divisors D1 = {L2,L3} and D2 = {L1 ,L4}. We next construct a polyhedron n1 = n1 + iR2 compatible with this set of divisors, with the skeleton 7 + iR2. Fig. 3 shows a two-dimensional cone (sector) n1 C R2 generated by rays which are parallel to the real sections of L3 and L4. It forms the polyhedron n1. Second, we consider divisors D[ = {L3, L4} and D2 = {L2}. A cone n2 generated by rays which are parallel to the real sections of L3 and L2 forms a polyhedron n2 = n2 + iR2, compatible with the set of divisors D1, D2, see Fig. 4.

We can see in Fig. 3 that families L5,L6 as well as Li,L2, LI, L4 come into the polyhedron ni, so in the cone ni there are points at which two, three and even four lines intersect. However, the form Q has nonzero residues only at points z(k) = (zi(k), z2(k)) with coordinates

18 4

Zl(k) = 3 + 3 kl - 3 :

14 8 z2(k) = 3 - 3ki + 3k2, k = (k1,k2) G Z|.

(13)

The intersection points (13) of lines L3, L4 are indicated in Fig. 3 by a black colour. Hence, the sum of local residues at points z(k) yields the Puiseux series

Pi(x) = —L^ E x-8'3kl+4'3k2x^1-8^2 (14)

xi x2 kezl.

with coefficients

c=(-i) 1 k I + I ki - 4 k2)r( i - 3 ki + I k2) 1

ck = k! r( I + 1 ki - 4 k2)r( 4 - I ki + I k2) 9(1 - 4ki - ^ (15)

Four families of lines L2, L3, L4 and L5 come into the polyhedron n2, see Fig. 2. However, the form Q has nonzero residues only at points z(k) = (zi(k), z2(k)) with coordinates

zi(k) = l+2ki + 1 k2, (i6) z2(k) = -k2, k = (ki,k2) G Z|.

Points (16) are black in Fig. 4, where lines L2, LI intersect. The sum of residues at z(k) yields the Puiseux series

P2(x) = At2 E Ckx-2kl-i/2k2xk2 (17)

x

i/2

1 kez%

with coefficients

k = ^ r ((23 +2k1 + f k2) r/71 - f k1 + k2) 1(1 - 4ki + k2). (18)

k k! T(3 + ki + 2k^T(7 - 1 ki - Ik2) 16K ' y '

We remark that arguments of T-functions in coefficients of the series (8) and also in (15) and (18) can be real nonpositive numbers, which are poles for the function r. So, by a ration of two r-functions we mean a meromorphic function with removable singularities at those points. For instance, we mean

r(-i) = r(-1) = 1 r(0) -r(-i) ■

So, series (14) and (17) are analytic extensions of the series (8).

We now characterize domains of convergence of Puiseux series obtained above in the logarithmic scale. According to the two-sided Abel lemma for hypergeometric series [10], there exists a relationship between the structure of the convergence domain of this series and its support. Since series (14) and (17) represent branches of the multivalued algebraic function yi(x) ■ y2(x) with singularities on the discriminant set of the system (7), projections of convergence domains of such series on the space of variables log |xi|, log |x2| are unions of several components of the discriminant amoeba complement, see Fig. 1. We recall that the amoeba of the algebraic set

V c Cn is defined to be the image of V under the mapping Log. In this way, the series (14) converges in the domain G\ = Log-1(E3), where E3 is an amoeba complement component. The projection Log(G2) of the convergence domain G2 of the series (17) covers two components E1, E2 and an amoeba tentacle separating them, see Fig. 1.

Fig. 3. The real section of polar divisors. The Fig. 4. The real section of polar divisors. The

cone

cone

2. Taylor series for monomials of solutions to reduced systems

We consider the system of n trinomials (1) with unknowns y1,... ,yn, variable coefficients a =(..., a(\ ...) and the set of supports A(l\ ..., A(n), the same as the system (2) has.

Let us denote by A the disjunctive union of sets A« It consists of 3n elements, and we interpret it as the (n x 3n) - matrix

A = = (a\...,a3n),

with columns ak which are exponents of monomials of the system (1). We order elements a G A, and, correspondingly, coefficients a(i), a G A of the system (1). The set of coefficients a = (aa) is a vector space CA ~ C3n.

The system (1) can be reduced by an appropriate change of coefficients in such a way that only one variable coefficient remains in each equation, and the other ones will be constant as in the system (2). Herein, supports A(1),..., A(n) remain the same, and the solution to the system (1) can be restored by the solution to any reduced system. On the whole, the reduction procedure (dehomogenization) of the system is based on the polyhomogeniety property of the solution y(a) = (y1 (a),... ,yn(a)), which can be expressed as follows:

y(... X0i)Xaa(i) ..) = (x-^a^.),X-^a^..)), (19)

where A0 = (a01) ,..., X^, X = (X1,...,Xn) G (C \ 0)n, see [4].

In each set A(i) we fix a pair of elements ¡(i), v(i) and form the n x n-matrix

k := (¡(1) - v (i),...,M(n) - v(n)) (20)

with columns ¡(i) - v(i). The matrix k is assumed to be nondegenerate. Each fixed set of n pairs ¡(i), v(i) corresponds to the reduced system of trinomials

r% yT + - =0, i =1,...,n, (21)

with new unknown y = (yi,... ,yn), variable coefficients r = (r^^ G Cn and /(i) G A(i). In

each set A(i), we can choose an unordered pair ¡(i),v(i) in three ways. Hence, we consider at most 3n ways of the reduction of the system (1) to the form (21). If ¡(i) = w(i), v(i) = 0 and ¡8(i) = a(i) for all i G {1,... ,n}, then we get the system (2).

Consider a branch of the solution to the system (21) under condition yi(0) = 1 and call it the principle solution. For the vector d = (di,..., dn) G R+ we introduce the monomial function yd(r) := yd, (r) ■... ■ y^ (r) of coordinates of the principal solution to the system (21). Concerning the system (21), we use the following notations: /3 is the matrix formed of columns /(i) and / is the matrix with columns (/(i) - v(i)). Moreover, the symbol T(b) we will use for the short

n

writing of the product H r (bk), where b = (bi,... ,bn) is a vector. The diagonal matrix with

k = i

components of the vector b on the main diagonal we denote by diag[b] and the I denotes the vector with unit coordinates.

Theorem 1. The monomial yd(r) of the principle solution to the system (21) admits the Taylor series representation with coefficients

(-1) I k I r(K-id + K-i/k)

Ck = -T--f-7--7-)Q(k), k G Zl, (22)

k k! Y{x-id + K-i/3k - k +1) W l V 7

where Q(k) is the determinant of the matrix (diag [K-id + K-i/k] - k-i/ diag [k]), k! := ki! ■ ... ■ kn! and I k |:= ki + ... + kn.

Proof. Following [4], we carry out the linearization of the system (21). For that we regard (21) as a system of equations in the space Cn x CJJ with coordinates r = (r^^), y = (yi,..., yn), and introduce in Cn x Cn the change of variables (£, W) ^ (r, y) by setting

y = W-K-1, r = £ © WK-1fi-E, (23)

where £ = (£i,..., £n), W = (Wi,..., Wn), © denotes the Hadamard (coordinate-wise) product and E is the unit matrix. As a result of this change of variables, the system (21) can be written in the vector form as follows

W = £ + I. (24)

Equations of the system (24) are linear, so the change of variables (23) is called the linearization. Coordinates of the solution to the system (21) in new variables £ = (£i,... ,£n), W = (Wi,..., Wn) take the form

yj (r(£)) = (Wi,...,Wn)-(K ) , where Wi = 1 + £i, (k-1 )(j) is the jth column of the inverse matrix k-1 for the matrix k.

We represent the inversion £(r) of the linearization (23) as an implicit mapping given by the following set of equations

F (£, r) = (F1 (£, r),..., Fn(£, r))= £ © W K-1fi-E - r = 0. (25)

Calculate the vector y(£) at the value of the mapping £(r). To this end, following the idea implemented in [8] for a system of polynomials with a diagonal matrix w, we apply the logarithmic residue formula, see [5, Th. 20.1, 20.2]. It yields the following integral

yd(r) = f ydm(№

y () (2ni)n J F(£,r) ,

rE

where Te = {£ G Cn : |£j| = e, j = 1,... ,n}, A(£) is the Jacobian of the mapping (25) with respect to £ and F(£,r) denotes the product F1(£,r) ■ ... ■ Fn(£,r). The radius e we choose in such a way that the corresponding polycylinder lies outside the zero set of the Jacobian A(£).

Lemma 1. The Jacobian of the mapping F(£,r) with respect to £ is

A(£) = W(K-1?)I-21 det (E + (\vAg[£]H-1p) .

Proof. The j th component of the mapping F (£,r) has the following form:

n 1

Fj = Fj (£, r) = j n Wt-1"-E)i3) - j. k=1

The calculation of the derivative of Fj with respect to £j looks as follows:

dFj IT uA*-1?-E)i3) , c t 1 a m(j) IT uAk-1

dj = 11 wkK >-E)k + j(K-1a - E)jj^n W

£j k=1 k=1 n

(1+j (K-j n Wr1 k=1

and the derivative with respect to £i, when i = j, is equal to

dF = £j (k-1 /a - E)j n W(K-1 -', °ei k=1

where S3k, S\ denote the Kronecker symbols.

Extracting common factors in the rows and columns of the obtained determinant, we get the assertion of the lemma. □

Remark that at the origin the Jacobi matrix for the mapping F(£, r) is the unit matrix. Hence, the Jacobian A(£) does not vanish in the neighborhood of the origin and conditions of Theorems 20.1, 20.2 from [5] hold.

The monomial yd(r) after the change of variables takes the following form:

yd(£) = W-K-1d.

Consequently, application of the logarithmic residue formula yields the integral representation:

1 r W-K-1d+(K-1 ¡3)I-2I

yd(r) = Jnfj-Her-^(E+diag[e]K-1P) d£. (26)

rE

Expand the kernel of the integral (26) into a multiple geometric series. To this end, we use the coordinate notations:

yd (r)

1

W-K d+(K-1 /3)1-21

(2ni)n

(2ni)n

ft(. ,=1 w-K- , n wr1 k=1 ad+(K-1 /3)I-2I

n (

e, (1 ^-^^jr

,=1

\ fc=1

W-K-1d-1

-det (E + diag^K-1 ¡3) d£

■det (E + diag[^]K-1/3) d£

-iR-^k

(2ni)n

det (E + diag[^]K-1 /3) d£.

ne, 11-

Since there exists such a number S that for all £ € Te and ||r|| < S the inequality

r< < 1

e, n wr

k = 1

)(j)

is valid, the integral (26) admits the following representation: 1 f W-K-1d-1 det (E + diag[£]x-1 /3)

y (r) =

(2ni)n

ne,

,=i

^ n ( e,w(K-1 ¡}-E)(j)) )

de.

Changing the order of summation and integration in the last integral, we get the series

1 f W-K-1(d+j3k) + k-I

yd(r) = ]T ( J^—yn I -rt+7-det (E + diag[£]x-13) d£ | rk.

keZ'i

(2ni)n

ek+i

The coefficient ck of the series is determined by the expression in parentheses. It can be calculated by the Cauchy integral formula. As a result, we get:

ck = 1 dk (w-K-1(d+W+k-1 det (E + diag[e]K-1 /})

«=0

We bring the factor W K 1 (d+@k)+k 1 into the determinant in such a way that each row of it still to depend on one variable j. We obtain

Ck =

1 dk k!

det(diag Wdiag[-K 1(d+^k)+k-l]] x (E + diag[e]K-1/3)j

«=0

We next use the multilinearity property of the determinant and the fact that each row depends only on one variable £j. As a result, we have

Ck = 1 det

dkj (-K-1(d+f3k)), + kj-1

—r W1

dek

(s1+e, (k-1,

«j =0

i,, = 1

)

r

,

1

K

1

r -i

(K-1R-E)(P

,=1 \ «j n Wk -

k = 1

n

Finally, we perform calculations in the above determinant:

^ W^ ^^- (ôj + j (K-i j

at-kj" j \Wi 1

3 =0

kj-1

(-lp((x-1(d + _k)). ôj - kj (x-1H)f)n ((K-1(d + m)j - kj + m)

m=1

r f (k- 1 (d + _k))) , V-_ ) j ) i(K-1(d + &)), ôj - kj (K-1 j .

-1( J I 7„ I 1 V j /

^(K-1(d +_k)) . - kj + l)

Taking out the common factor in each row of the determinant and taking into account the factor

77, we get the view of the coefficient ck declared in formula (22). □

k!

Coefficients of the Taylor series for the monomial yd(x) of the principal solution to the system (2) one can find by formula (22) setting k = w, 3 = a. Thus, the series is as follows:

^x) = E (-T\f rW1d \n Pk)xk, (27)

y K ' ^ k! r(w-1d + w-1ak - k +1) w ' v 7

kezj v '

where P (k) = det (diag [w 1d + w 1 ak] - w 1a diag

3. Puiseux series

We fix n couples p(i), v(i) G A(i) of exponents of the system (2) and compose the matrix

( (i)) ( (i) (i)) K = H ) = № - v)),

assuming that it is nondegenerate. In accordance with the choice of the set of pairs p(i), v(i), let us devide the set {1, . . . , n} on three disjoint subsets:

J = {j : v(j) =0, p(j) = w(j)},

L = {l : v(l) =0, p(l) = a(l)}, (28)

T = {t : v(t) = a(t), p(t) = w(t)}.

We introduce two matrices

$ := k-1 ■ a, ^ := k-1 ■ w, with rows ... ,yn and ^1,... respectively. Moreover, we consider truncated rows

rf, tf, tf, l G L, , tf, ^T, t G T,

which consist of entries of rows l G L and y>t, t G T indexed by elements of sets J, L

and T. Respectively, we introduce truncated vectors kJ, kL, kT for the vector k = (k1,..., kn). The scalar product of vectors we denote as follows (■, ■).

Theorem 2. For any collection of n couples ¡j(i) , v(i) € A(i) with the nondegeneracy condition of the corresponding matrix k there exists an analytic continuation of the Taylor series for the monomial yd(x) of the principal solution to the system (2) in the form of the Puiseux series

E ¿kxm(k),

kezj

which has the support consisting of points m(k) = (m1(k),..., mn(k)) with coordinates mj (k) = kj, j € J.,

ml(k) = -{<pJ, kJ) - {tf, kL) + {tf, kT) - {d, k-1), l € L, mt(k) = {VJ,kJ) + {^L,kL) - {^T,kT) + {d, k-1), t € T, and coefficients Ck expressed in terms of coefficients (22) as follows

in (kt+mt(k))

Ck = e teT Ck.

Proof. We start the proof with finding the monomial change of variables r = r(a) reducing the system (1) to the form (21). To this end, we get the Smith normal form Sq for the matrix k, multiplying it on the left and right by unimodular matrices C and F as follows:

C kF = Sq, (29)

here the Sq is a diagonal matrix with integers q1,...,qn on the diagonal, and qj | qj+1; 1 < j < n - 1, see [11]. It follows from (29) that the inverse matrix k-1 admits the representation

x-1

= FS-1 C. (30)

As it was mentioned above, the solution y(a) of the system (1) is polyhomogeneous. We find the polyhomogeneity parameters A0j) and A = (A1;..., An) such that

A^V' a« = 1, A^A^ a^li) = -1, for i = 1,... ,n. For that, we solve the following system of equations:

(31)

AkW = gi, i = l,...,n, (32)

where

(i)

a (i)

9. =__viii

9i = (i) .

Using the relation (30), we can write the solution of the system (32) in the matrix form as follows

___ s C nA qn 1

/ 1 \ c

a=g-1=gFs-ic = (g(i))qi,...,g(n)),

where the vector g has coordinates gi, and f(1),...,f(n) are columns of the matrix F. By

i

choosing for each i all qi values of the radical (gf q, we yield all branchers of the matrix radical gK 1. There are \ det k \ = q1 ■ ... ■ qn of them.

For each i G {l,... ,n} we find the parameter Ag , using one of relations (31). If vi(i) = 0, then Ag^ = —t— . For ¡j,(i) = w(i) we get

(i) ' ay

, / i iN -Cw(i)

Ai) 1 f f (i)\ Il f f qn \

Ao = an, \K ) ,--Agf ) )

If i G J, then the coefficient r^ of the system (21) can be expressed in terms of coefficients a of the system (1) in two ways:

Ai) , ) <ci,v(i)) , ) (cn,v(i)>

r% = - if • (gf (1)) qi • • g(n)) qn ,

a

0 (33)

(i) <ci,^(i)-u(i)) <cn,y(i)-u(i)) V '

= Oir • (nf(1)) qi • • (nf(n)) qn •

' ad) ~ (i)

a '(i)

If i G L, then the coefficient r^(i) of the system (21) can be expressed in terms of coefficients a of the system (1) as follows

a(i) , ) <ci^(i)) ) <cn,»(i))

r% = - OH • (gf (1)) qi • • (gf (n)) qn (34)

For i e T the relation is as follows

a(i) , ) <ci ,^(i)) ) (c„,*(i))

(i) = OL. nf (i)\ " . .(nf(n)

(gf(y•... • (f (n)Y. (35)

r0o = — • (n ) •... •

au(i)

In formulae (33)-(35) vectors c1,... ,cn are rows of the matrix C.

In particular, if for all i G {1,..., n} we choose p(i) = w(i), v(i) = 0, then L = 0, T = 0 and k = w. The matrix w is nondegenerate by assumption and the system (21) coincides with the system (2). In this case, we get the change of variables x = x(a). It can be written in two ways:

a(i) , ) <ui,v(i)) , ) <un,v(i))

Xi = - Of • (hv(i)) Pi • ... • (hv(n)) pn ,

ao (36)

(i) <ui,^(i)-v(i)) <un,&(i) -^(i))

Xi = Of •(H")-^— :...(„")

Pn

(i) a0

In formulae (36) the vector h has coordinates hi = —(0—, vectors u1,... ,un are rows of the

a ii)

unimodular matrix U, in turn, vectors v(1),..., v(n) are columns of the unimodular matrix V such that w = USpV, where Sp = diag[p1,... ,pn], Pj | Pj+1, 1 ^ j ^ n - 1.

(i) ao.

Remark that gi = hi for i G J. Furthermore, if i G L then gi =--, and for i G T we have

a%)

(i)

a i)

gi = —(i) . Getting these ratios from (36), we substitute the expressions for gi into (33)-(35). a ¡i)

As a result, we get coordinates of the monomial transformation r = r(x) for the transition from the system (21) to the system (2):

(j) = x.TT-<3).TT (_x.)Vt]

ra(3) = xj I IX

IK* -n -xtf * G J

leL teT

r(j) =n T[ -xt/" ,3 G L, (37)

leL teT

rj = - n xf - n (-Xt,)-^ ,3 G T.

leL teT

According to the polyhomogeneity property (19), the division of the jth coordinate of the solution to the system (1) on Xj = 0 is compensated by the multiplication of the coefficient a<0i) on Xa. So taking into account (32) we obtain the relationship between monomials yd(x) and yd(r) of the following form:

> 3) d/„\ _ TT gj_„ d

M3)

, K -g 3

y-(x) = n yd(r^ (38)

j=l hj

where k- 1, are jth rows of matrices k-1 and w-1 correspondingly. Using relations (36), and the fact that gj = hj for j G J, we write (38) as follows:

yd(x) = n x-{d'Kjl) n xt)(d' Kjl)yd(r). (39)

leL teT

Hence, making the substitution (37) in the expansion (22) and taking into account the relation (39), we conclude, that the support S of the required Puiseux series consists of points m(k) = = (m1(k),..., mn(k)) with coordinates

mj (k) = kj, 3 G J,

mi(k) = -(f J, kJ) - &L, kL) + &T, kT) - (d, k-1), l G L, mt(k) = (fJ,kJ) + (rt,kL) - №,kT) + (d, k-1), t G T.

The coefficient ck of the Puiseux series is expressed in terms of the coefficient (22) by the following formula

in J2 (kt+mt(k)) Ck = e teT Ck.

□

As mentioned in Section 1, by the two-sided Abel lemma for hypergeometric series [10] the cone of the support S of the series defines the logarithmic image Log(G) of the convergence domain G of the series. This means that the geometry of the domain G is closely related to the structure of the amoeba A of the discriminant hypersurface V of the system (2). The amoeba A can be obtained from the amoeba A of the discriminant set of the system (21) via an affine transform associated with the change of variables r = r(x). Consequently, the recession cone of the set Log(G) for the Puiseux series of the monomial yd(x) is the image of the negative orthant -R+ under an affine transform.

In conclusion, we return to the example from Section 1 to make the following remark. By Theorem 2 we associate the Puiseux series (14) with couples of exponents:

(2, 1), (0, 0) G A(1), (1, 2), (0, 0) G A(2),

and, accordingly, the series (17) with the set

(2,1), (0,0) G A(1), (0,4), (0,0) G A(2).

The first author was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS"(no. 18-1-7-60-1). The second author was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS"(no. 18-1-7-60-2). The third author was supported by the grant of the Ministry of Education and Science of the Russian Federation (no. 1.2604.2017/PCh).

References

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Аналитические продолжения решений систем триномиальных алгебраических уравнений

Ирина А. Антипова Екатерина А. Клешкова Владимир Р. Куликов

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Статья посвящена исследованию аналитических продолжений мономиальной функции координат решения приведенной триномиальной алгебраической системы. В частности, показано, как техника интегральных представлений Меллина-Барнса и свойство полиоднородности решения универсальной триномиальной системы применяются для разрешения задачи аналитического продолжения. Таким образом, получены разложения Пюизо (с центром в нуле), представляющие аналитические продолжения ряда Тейлора указанной мономиальной функции.

Ключевые слова: алгебраическое уравнение, аналитическое продолжение, ряд Пюизо, дискриминант, интеграл Меллина-Барнса.