ON TRANSCENDENTAL SYSTEMS OF EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-3-326-343 УДК 517.55

On Transcendental Systems of Equations

Alexander M. Kytmanov* Olga V. Khodosf

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.12.2020, received in revised form 22.01.2021, accepted 20.03.2021

Abstract. Several types of transcendental systems of equations are considered: the simplest ones, special, and general. Since the number of roots of such systems, as a rule, is infinite, it is necessary to study power sums of the roots of negative degree. Formulas for finding residue integrals, their relation to power sums of a negative degree of roots and their relation to residue integrals (multidimensional analogs of Waring's formulas) are obtained. Various examples of transcendental systems of equations and calculation of multidimensional numerical series are given.

Keywords: transcendental systems of equations, power sums of roots, residue integral.

Citation: A.M. Kytmanov, O.V. Khodos, On Transcendental Systems of Equations, J. Sib. Fed. Univ. Math. Phys., 2021, 14(3), 326-343. DOI: 10.17516/1997-1397-2021-14-3-326-343.

Introduction

Based on the multidimensional logarithmic residue, for systems of non-linear algebraic equations in Cn formulas for finding power sums of the roots of a system without calculating the roots themselves were earlier obtained (see [1-3]). For different types of systems such formulas have different forms. Based on this, a new method for the study of systems of algebraic equations in Cn have been constructed. It arose in the work of L. A. Aizenberg [1], its development was continued in monographs [2-4]. The main idea is to find power sums of roots of systems (for positive powers) and then, to use one-dimensional or multidimensional recurrent Newton formulas (see [5]). Unlike the classical method of elimination, it is less labor-intensive and does not increase the multiplicity of roots. It is based on the formula (see [1]) for a sum of the values of an arbitrary polynomial in the roots of a given systems of algebraic equations without finding the roots themselves.

For systems of transcendental equations, formulas for the sum of the values of the roots of the system, as a rule, cannot be obtained, since the number of roots of a system can be infinite and a series of coordinates of such roots can be diverging. Nevertheless, such transcendental systems of equations may very well arise, for example, in the problems of chemical kinetics [6,7]. Thus, this is an important task to consider such systems.

In the works [8-21] power sums of roots are considered for a negative power for different systems of non-algebraic (transcendental) equations. To compute these power sums, a residue integral is used, the integration is carried out over skeletons of polycircles centered at the origin. Note that this residue integral is not, generally speaking, a multidimensional logarithmic residue or a Grothendieck residue. For various types of lower homogeneous systems of functions included in the system, formulas are given for finding residue integrals, their relationship with power sums of the roots of the system to a negative degree are established.

* AKytmanov@sfu-kras.ru https://orcid.org/0000-0002-7394-1480 1 khodos_o@mail.ru © Siberian Federal University. All rights reserved

The paper [12] investigated more complex systems in which the lower homogeneous parts are decomposed into linear factors and integration cycles in residue integrals are constructed from these factors. In [11], a system is studied that arises in the Zel'dovich-Semenov model (see [6,7]) in chemical kinetics.

The object of this study is transcendental systems of equations in which the lower homogeneous parts of the functions included in the system form a non-degenerate system of algebraic equations: formulas are found for calculating the residue integrals, power sums of roots for a negative power, their relationship with the residue integrals are established. See [21].

1. The simplest transcendental systems of equations

Consider a system of functions of the form

fi(z), f2(z),...,fri(z),

holomorphic in a neighborhood of the point 0 € Cn, z = (zi, z2,..., zn) and having the following form:

fj (z) = zß + Qj (z), j = 1, 2

(1)

where ¡3j = (Pj,... ,P°n) is a multi-index with integer non-negative coordinates, zn = zn ■

z22 ■ ■ ■ zn and || = ¡3\ + ¡32 +... + @n = kj, j = 1, 2,.. .,n. The functions Qj can be expanded in absolutely and uniformly converging Taylor series in a neighborhood of the origin of the form

Qj (z)= E

(2)

11 a y j

where a = (ai, a2,..., an), aj ^ 0, aj G Z, a za = za1 • z^á2

Consider the cycles y (r) = Y(r 1,r2,..., rn), which are skeletons of polydisks: Y (r) = {z £ Cn : \zs\ = rs, s = 1, 2,...,n}, r 1> 0,...,rn > 0. For sufficiently small rj, the cycles y(r) lie in the domain of holomorphy of functions fj,

therefore the series

E \aá\rai••• ran, j = i,---n.

\\a\\>kj

converge. Then on the cycle y(tr) = Y(tri,tr2,... ,trn) for sufficiently small t > 0 we have

\zf = tk! • rß1 • rß2 ••• rj = tk! • rßj

and

\Qj (z) \ =

E -

\a\\>kj

aaz

<

< E tIIaIIK\ra < tk+1 E KK, j = !,..■

II a\\ || a||

Therefore, for such t on the cycle Y(tr) the inequalities hold

\zf > \Qj (z)\, j = 1, 2,...,n.

Thus

fj(z)=0 on y(tr), j = 1, 2,... ,n.

zrn

n

In what follows, we will assume that t =1. Consider a system of equations of the form

7i(z) =0, /2(z)=0,

[7n(z) =0.

From (3) it follows that for sufficiently small rj the following integrals are defined

(4)

1

df

J zß+I f J

Y(r) Y(ri,r2,

x)

_1_ dfl A f A A dfn

zf1 + 1- z^.. .znn+1 ' fl A f2 A ••• A fn '

where ¡31 > 0, f32 > 0,..., @n > 0, ¡3j G Z, I = (1,1,..., 1). We call them residue integrals ( [22]).

The logarithmic residue theorem does not apply to these integrals, and they are not standard Grothendieck residues.

Since condition (3) is satisfied on the cycles 7(r), by the Cauchy-Poincare theorem, these integrals are independent of (r1,... ,rn). Let us denote

Jß

1

1

_ _ df

(2ni)n J zß+r f •

Y(r)

Theorem 1. Under the assumptions made, for a function fj of the form (1), (2) the next formulas are valid

J8 =

E

(-1)"

Il a II ^ II ß" +min(n,fci +... + kn )

dk(A ' Qa)

(ß +(ai + 1)ß1 + ••• + (an + 1)ßn)!

' dzß+(ai + 1)ß1+... + (an+1)ßn

]T (-1)IIaIIM

11 a I ^ II ß I +min(n,ki + ... + kn)

z=0

A ' Qc

zß+(ai + 1)ß1 + ... + (an + 1)ßn

where k = \\ß +(a1 + 1)ß1 + ••• + (an + 1)ßn\\, ß! = ß1! ' ß2! — ßn !, Qa = Q'í1 ' Q^2

' Q

an n

dm

dIIßI

dzß dzß1 dzß2 ''' dzß

A is the Jacobian of the system of functions (1) and, finally, M is

a linear functional assigning to the Laurent series (under the sign of the functional M) its free term.

Corollary 1. If all ßj = (0,0,...,0), j = 1,... ,n, then the integral

Jm

(-1)I

IHKIIßII

m

A ' Qa

E -rdm(AQ)

IMKIIßII

ß! dzß

z=0

Our further goal is to relate the considered integrals to power sums of roots of the system (4). To do this, we will narrow the function class fj. First, we take as functions Qj (j = 1, 2,..., n) polynomials of the form

Qj(z) = E <za, (5)

aeMj

a

p

where Mj is a finite set of multi-indices such that for a £ Mj the coordinates ak < ¡¡k, k = 1, 2,... ,n, k = j. (But it is still assumed that \\a\\ > kj for all a £ Mj).

Denote

M

&ß+I = &{ßi + l,ß2 + l,..,ßn+l) = E

ßl + 1 zß2 + 1 k=l zl(k) • z2(k)

zn(k)

where ¡3 = (¡1,..., ¡n) is some multi-index. This expression is a power sum of roots that do not lie on the coordinate planes of the system (4), but in negative power (or a power sum of the reciprocal of the roots).

Theorem 2. For the system (4) with functions fj of the form (1) and polynomials Qj of the form (5) the next formulas are valid

Jn = (-1)n^n+i,

i.e.

°ß+i = E (-1)

\\ a \\ ^ \\ ß\\ +min(n,ki + ... + kn )

a\ +n

M

A • Qc

yß+(ai+l)ß1 + ... + (an + l)ßn

Consider a system of equations in three complex variables

/l (zi,z2,z3) = 1 + a\z\ = 0, /2 (zi,Z2,Z3) = 1 + bizi + b2z2 = 0,

/3(zi, z2, z3) = 1 + cizi + C2z2 + C3z3 = 0.

(6)

Here the functions do not satisfy the conditions of Theorem 2, but they satisfy the conditions of Theorem 1. We find the integral

J(

1

1

(ß,0,0)

dfi A df 2 A df

(2ni)3 J zß1+lz2z3

Y(r)

fl • f2 • f3

aib2Csdzi A dz2 A dz3

(2ni)3 J zß1 + lz2z3 (1 + aizi)(1 + bizi + b2z2)(1 + cizi + C2z2 + C3z3)

Y(r)

aib2C3 dß

ß! dzß

1

(1 + a 1 z 1 )(1 + b 1 z 1 )(1 + c 1z 1) To calculate the last derivative, we transform the expression

1 A B

+

z1 = 0

+

C

(1 + aizi)(1 + bizi)(1 + cizi) 1 + aizi 1 + bizi 1 + cizi'

A

(ai - bi)(ai - Ci)

B = -

C=

b2i

(ai - bi)(bi - ci)''

(ai - ci)(bi - ci)'

assuming that ai = bi , ai = ci, bi = ci , then

J(ß,0,0) = (-1)ß aib2c3X

1

1

1

2

a

i

c

i

,ß+2

bß+2

„ß+2

+

(ai - bi)(ai - ci) (ai - bi)(bi - ci) (ai - ci)(bi - ci)

The roots of the system (6)

zi =

a1

Z2 =

bi - ai ai 62

Z3 =

62ci - bic2 + aic2 - ai62

aib2c3

If the numerator in the formula for z3 is 0, then this root lies on a coordinate plane, and we should not take it into consideration. Therefore, the power sum

(-ir+14+3bic3

(bi - ai)(b2ci - bic2 + aic2 - aib2) '

i.e.

J

(-l)ß a2b2c3bß+i

+

(bi - ai)(b2ci - bic2 + aic2 - aib2)

(-1) ß+iaib2c2c3

_____ X

(b2ci - bic2 + aic2 - aib2)

+

aic2 •

aß+i - cß+i ai - ci

+ (bic2 - b2ci)

b

ß+i „ß+i

bi - ci

(8)

We recall the well-known expansions of the sine into an infinite product and the power series:

sin v z

n (1

k=l

(-1)S

k2n2) ^ (2k + 1)!'

k=Q v '

which converge uniformly and absolutely on the complex plane. Consider the system of equations

fi(zi,z2,z3) =

/2 (zi, Z2, Z3) =

f3(zi,Z2,Z3) =

sin %/zi - a2

a/z1 - a2 sin %/z2 - zi -

n

k=l

a/z2 - zi - a2

1

n

m=l

zi - a

k2n2

1

z2 - zi - a

sin Vz3 - z2 -

2

Vz3 - z2 - a2

1-

s = l

z3 - z2 - a

= 0, 0.

(9)

Each of the functions of this system can be expanded into an infinite product of functions from system (6).

The roots of the system (9) are the points (n2h2 +a2, n2(h2+m2)+2a2, n2(h2+m2 + s2) + 3a2), k,m,s £ N. Therefore, the power sum /3+1,1,1) is equal to the sum of the series

~ 1

(n2k2 + a2)( 3+1)(n2(k2 + m2)

k,m,s=1

a(ß+l'l'l) (n2k2 + a2)(ß+l)(n2(k2 + m2) + 2a2)(n2(k2 + m2 + s2)+3a2),

which converges as a = nki. For the system (9)

fi = E

^ k

k=Q

(-1)k(zi - a2) (2k +1)! '

i

i

x

1

i

i

X

z

z

0

a

m2n2

s2n2

h = g-1)k(z2 - zi - a2)k

k=0

f3 = J2

k=0

(2k + 1)!

(-1)k(z3 - z2 - a2)k (2k + 1)! ,

therefore

— "2k sha

fi(0,00,00) = f2(0,00,00) = f3(0, 0, 0) = Y; T^+^y

k=0 ( + )!

Therefore, to apply the formula from Theorem 1, we need to divide the functions f1,f2, f3 by these constants (normalize).

Consider the integral J(n,0,0) for the system (9). Using the form of the roots of the system (9), we obtain that

1

ai =

bi =

1

n2k2 + a2 n2 m2 + a2

b = 1 = 1 =

b2 2 2,2 , c2 2 2,2 , c3

1

n2m2 + a2 '

n2s2 + a2 '

n2s2 + a2

E

k,m,s=i

J(ß,0,0) = -a(ß+i,i,i) + (-1)ß+1 x 1

(m2n2 + a2)ß+i(n2(k2 + m2) + 2a2 )(n2 (k2 + m2 + s2) + 3a2)

+

+(-1)ß+i V _1_

(n2s2 + a2)(n2(k2 + m2 + s2) + 3a2)

k,m,s=i

+

(-1)ß

(m2n2 + a2)ß+i (k2n2 + a2)ß+i

J(

(ß,0,0)

E

k,m,s = i

(n2 (k2 + m2) + 2a2)(n2(k2 + m2 + s2) + 3a2)

1

+

(-1)ß+i

(k2n2 + a2)ß+i (m2n2 + a2)ß+i_

+

+(-1)ß+i V _1_

(n2s2 + a2)(n2(k2 + m2 + s2) + 3a2)

k,m,s=i

+

(-1)ß

(m2n2 + a2)ß+i (k2n2 + a2)ß+i

For odd ¡3 the integral J(n,0,0) = 0, and for even ¡3 = 2n we obtain the following formula for finding the sum of the series

J(

(2n,0,0)

E m

\\a\\<2n

A • Qa

= —2a,

(2n+1,1,1)

-2

k,m,s=i

(n2k2 + a2)(2n+i) • (n2s2 + a2) • (n2(k2 + m2 + s2) + 3a2) '

Let us calculate, for example:

J{0,0,0) = M[A] = M Applying the identity

f. dh. f

dzi dz2 dz3

I —r---ctha

V2a2 2a

+

(n2k2 + a2)(n2(k2 + m2) + 2a2) (n2m2 + a2)(n2(k2 + m2) + 2a2)

a

x

1

x

1

X

X

1

X

X

2

n

1

1

3

1

1

(n2k2 + a2)(n2m2 + a2) '

We get that

2a(i,i,i) = E

Thus, we get

1

(n2k2 + a2) • (n2s2 + a2) • (n2(k2 + m2 + s2) + 3a2) '

V _i_) =

(n2k2 + a2) • (n2(k2 + m2) + 2a2) • (n2(k2 + m2 + s2) + 3a2)7

k ,rm,s — 1

(actha — 1)3 = 48a6 '

2. Special systems of equations

Consider a system of functions f1(z), f2(z),- - -, fn(z) of the form

'ji(z) = (1 — auz1)m11 ••• (1 — a1nzn )min + Q1(z), f2(z) = (1 — a21z1)m21 • • • (1 — a2„z„ + Q2(z),

Jn(z) = (1 — an1z1)mn1 ••• (1 — annzn)mnn + Qn(z),

(10)

where mij are natural numbers, aij are complex numbers that are different for fixed j, Qi(z) are entire functions, i = 1, - - - ,n. Let J = (j1, - - - ,jn) be a multi-index, where (j1 - - - jn) is a permutation of (1, - - -, n). Let us define aj = (a1j1, - - - ,anjn) for a multi-index J. We denote

qi(z1, zn) = (1 — ai1z1)m" ••• (1 — ainzn)min, i = 1,-",n, (11)

then the system (10) can be rewritten as

fi(z1, - - - ,zn) = qi(z1, - - - ,zn) + Qi(z1,-",zn), i = 1,''',n' (12)

For each m we define the function

{qm(z) 1 1 if amj = 0 for all j;

qm(z) • • ' ' ' • if amji ' ' ' aijk 0- ()

zjl zjk

A system

hm(z) = 0, i =1,--',n, (14)

has n! isolated roots in Cn, where C = C x • • • x C. Recall that C is a compactification of the complex plane C (the Riemann sphere). Then C is one of the well-known compactifications of Cn (the function theory space). The roots of the system (14) are equal

a f (1/a1ji,-'-, 1/anjn) if akjk =0 for k = 1,--',n,

\ (1/a1ji, - - - , OO [ii], - - ^ m[iк], - - - , 1/anjJ if aiiji1 = - - - = aik jik = ^

where k,j = 1, - - - ,n. If ajl il = 0, then in aJ we write o, this is the point at infinity in C.

1

By rh we denote the (global) cycle:

rh = {z G Cn : \hm\ = rm, n > 0, m = 1,..., n}. (15)

In the case when all ak,jk = 0, we define the (local) cycle rh,aj as follows

(16)

(17)

\1 - a1ji z1\ = r1, \1 - a2j2z2\ = r2,

\ 1 anjn zn \ rn .

If ailjii = ... = aikjik = 0 for some i1,..., ik, then rhi3iJ is defined as

'\1 - a1ji z1\ = ru \1/zii \ = rii, \1/zik \ = rik, \1 anjn zn \ rn .

Lemma 1. For sufficiently small rm, the global cycle rh has connected components (local cycles) in the neighborhood of the roots aj. Moreover, rh is homologous to the sum of local cycles rhi,aJ ■

Consider the system of equations

Fm(z,t) = qm(z) +1 ■ Qm(z)=0, m =1,...,n, (18)

depending on the real parameter t ^ 0. Let r1 > 0,... ,rn > 0 be fixed real numbers. Then, for sufficiently small t > 0, the inequalities

|qm(z)| > |t ■ Qm(z)\, m =1,...,n

on cycles

= {z £ Cn: \hm\ = rm, m =1,...,n}

because rh is compact.

We denote by JY(t) the residue integral

,,, 1 f 1 dF

JY(t)=(2nvnJ zy+i • F = (19)

dFi dF2 dFn

i A ——2 A —n

(2ni)n J zYl+1 ■ zY2+1 ■■■ znn+1 F1 F2 1 2

where y = (y1, ..., Yn) is a multi-index, and I = (1,1,..., 1).

We denote by A = A(t) the Jacobian of the system of functions F1(z,t),..., Fn(z,t) in the variables z1,... ,zn.

1

1

Theorem 3. Under the assumptions made on the functions Fi defined by formulas (18), the following expressions for JY(t) are absolutely convergent (for sufficiently small t) series:

, (-1)s(J^-^IMI + IIß(a,J)l+™

JY(t) = > ^^ X

J a

d IIß(a(J)II dz ß (a,J)

ß(a, J)! • aß+I

A(t) Qa

Yl + l

-,Yn+l qa+I

qa+I(J)

where ( — 1)s(J) is the parity of the permutation J, a = (a1, - - -, an) is a multi-index of length n, qa+T (J) = q^+j • • • qan+1[jn], and qs[js] is the product of all (1 — aj1z1)]i • • • (1 — a,jnzn)min, except (1 — a,Sjszs)msjs,

P(a, J) = (m1ji (aji + 1) — 1, - - -,mnjn (a^ + 1) — 1),

P(a, J)! = ]}(mpjp (ajp + 1) — 1)!,

p

a3+I = nmHi K'i +1) amnjn (ajn + 1)

aj = a1ji • • • anjn ,

Q\\P(a(J )\\ d]iji («¿i +1)-1 + ...+]nj„ {ain + 1)-1

dzl(a,J) = dz]iji (aji+ 1)-1 dz]njn (ajn + 1)-1'

The dash at the summation sign means that the summation is performed over all multi-indices J for which there are no zero coordinates in aJ.

Suppose Qs(z) are polynomials:

Qs(z) = zi ••• z^ Csaza s =1,...,n,

(20)

an >Q

where a is a multi-index, za = z^1 • • • zan, and degzj Qs ^ msj, s,j = 1,... ,n for all nonzero asj. If asj = 0, then there are no restrictions on the degree degzj Qs.

Assuming that all wj = 0, we make the change zj = —-, j = 1,... ,n in the functions

Fs(z,t) = (qs(z)+ t • Qs(z)), s = 1,...,n.

Hence, for s = 1,... ,n we have

F.s[ —,...,— ,t) = qs( —,...,—) +1 • qJ — ,...,—

Wl wn \wi wnl \wi wn

1

1 - asi— I • • • ( 1 - asn — w1

1

+1 • qJ—,..., —

w1 wn

w1

— ) "" • (wi - asi)ms1 • • • (wn - aSn)msn + t • QA —,...,— ) . wn w1 wn

Then we arrive at the formula

fA -L,...,-L,t) =

w1 wn w1

— ) • (qs(w)+ t • Qs(w) ) ,

(21)

where qs are functions

qs = (wi - asi)ms1 • • • (wn - aSn)

i

n

z=aj

m s i

msi

1

n

m

and Qi are polynomials

qs=wmsi ••• wmsn • —,..., — ).

i

i

Wl Wn, From the formula (20) we obtain

deg№j Qs < msj, s,j = 1,...,n.

We denote ___ ___ _

Fs = Fs(w,t) = qs(w)+ t • Qa(w), s = l,...,n. (22)

If 0 ^ t ^ l, then the system (22) has a finite number of roots in Cn that depend on t. Moreover, (22) has no infinite roots in C . Consider the cycle

rh = w G Cn :

i

i

W1 Wn

£s, S = 1, . . . ,n

for t close enough to zero. The compactness of the cycle rh implies

i

i

Wl wn

>

i

i

t • Qs — ,...,—

Wl Wn

S = 1,

Therefore, is homologous to the sum of cycles r^

1

alj1 — Wl = el,

1

a2j2- W2 = e2,

1

anjn Wn en

obtained from the cycles rh, j by replacing zj = —.

The equation

1 - ajsj — j Wj

defines a circle. Indeed, we rewrite it as

\Wj - ajSj \ = e\Wj \ or \Wj - ajsj\2 = £2\Wj I2,

then

or

(1 - e2)

1- £2

2 £2- a

(1 - £2

1- £2

2 e2 •\ajsj\2

, j = l,...,n,

(23)

(l - £2)2

for sufficiently small £ the point ajSj lies outside the circle and, therefore, rh,aJ is homologous to the cycle rh,aJ :

Wl - aiji \ = £1,

\W2 - a2j2 I = £2,

Here some acan be zero.

, \ Wn anjn \ en

j

e

2

a

js

W

j

a

js

Lemma 2. The residue integral (19) is

J Y (t) =

-I

(2ni)n

„Y1 + 1 „ ,Y'2+l

7 +-, dFl dF2 dFn

• Wln+l • W- A^2 A ... A-^

Fl F2 Fn

A-^J- A ... . (24)

Theorem 4. The following equalities are valid

p

E-

j=1 Zjl(t)Y1 +l • Zj2(t)Y2 + l ••• Zjn (t)Yn + l

J2(-t) l lKl l+nE

(-i)s(J) gl l ß(K,J)l

Keïï

ß (K,J)! dWß (K,J)

A(t) • w7

,71 + 1

Q

K

qK+I

(J )

Since zeros of (22) are polynomials in t, the equality (4) also holds for t = l. We denote

E

(Ty+i z71 + l z72 + l

j = l '

.,7n + l '

where z(j = (zji,.. .,Zjn) = (zji(l),Zjn(l)), j = 1,...,n.

Theorem 5. For the system (10) with functions fj defined in (12) and Qa defined in (20), the following formulas hold:

E-

i

(Ty+i z71+l z72 + 1 z7n + l

j = l Zjl ^ Zj2 ••• Zjn

(2ni)n

E (-i)l|K^T,(-i)'

(J)

llKll>0

A--

,71 + l

,,7n + l

Qk1 •... • Qb

ñk1 + l ñkn+l

qi • ... • qn

dW =

h,aj

v(-n I I Kl I (-i)s(J ) dllß(K'J )H

) J ß(K,J)! dWß(K,J)

A■ w^1+1

,,7n + l

Q

K

ñK+I

(J )

Consider the following system of equations in two complex variables:

\fi(zi,z2) = (i - a2Z2)2 + a^zizl = 0, \f2(zi,z2) = (i - bizi)2(i - b2Z2) + bsz2Z2 = 0.

(25)

Then Qm, m = l, 2 have the form (20). The system (25) has, as is easy to verify, 5 roots (zj1, zj2), j = l, 2, 3,4,5. If a2 = b2, then these roots do not lie on the coordinate planes.

Let us change the variables z1 = —, z2 = —. Our system will take the form

Wl W2

Jacobian of the system (26)

A

fi = wi(w2 - a,2)2 + a3 = 0, f2 = (wi - bi)2(w2 - b2) + b3 = 0.

(w2 - a2)2 2wi(w2 - a2) 2(wi - bi)(w2 - b2) (wi - bi)2

i

2

1

X

n

w=aj

1

jn

1

X

1

n

n

w = aj

= (w1 - bi)2(w2 - a2)2 - 4wi(wi - bi)(w2 - a2)(w2 - b2). Then, by Theorem 5, we obtain

= 5 ! 1 = (-i)\\K\\+sÜ)

= 1^ ' = ^1^ jn? x

j=i zji zj2 J KeK (2"l')

"Yl+1 ■ wY2+1 ■ al1 ■ bk2 ■ A ■ dw1 A dw2 -kl + 1(w2 - a,2)2(ki+1) ■ (wi - bi)2(k2+1)(w2 - b2)k^+1

„1-^^- a -.

I ki+1/ \2(k. +1) / 7 \2(k~ + 1W 7 \ k, + 1 V 1

fh„

Here the multi-indices K = {K = (ki,k2)\ 3m : Ym + 2 > ki + k2, m = 1,2}. The cycles rh,aj are cycles of the form {|wi| = rii, \w2 — b2\ = r22}, taken with positive orientation, and {\w2 — a2 \ = ri2, \wi — bi\ = r2i} are with negative orientation.

In particular, calculating J(o,o), after some transformations we obtain

(28)

(29)

A h a3b2 V(11) = 4a2bi - --^r^

(b2 — a2)2

without finding the roots.

Consider a system of equations in three complex variables:

fi(zi,z2, Z3) = 1 — aizi — a2Z2 — a3z3 + a1a2Z1Z2 + ai a3z1z3 + a2a3Z2Z3 = = (1 — a1z1)(l — a2Z2)(l — a3Z3) + a1a2a3Z1Z2Z3 = 0,

h(z\,Z2, Z3) = 1 — bizi — b2Z2 — b3Z3 + bib2ZiZ2 + b^zz + b2b3Z2Z3 = = (1 — bizi)(1 — b2Z2)(1 — b3Z3) + bib2b3ZiZ2Z3 = 0,

f3(zi,Z2, Z3) = 1 — CiZi — C2Z2 — C3Z3 + C1C2 Z1Z2 + C1C3Z1Z3 + C2C3Z2Z3 = ,= (1 — CiZi)(1 — C2Z2)(1 — C3Z3) + C1C2C3Z1Z2Z3 = 0.

The roots of the system (29) are (zji, Zj2, Zj3), j = 1,..., 12.

Change the variables zi = —, z2 = — and z3 = —. Our system will take the form

wi w2 w3

fi = W1W2 W3 — aiW2 W3 — a2Wi W3 — a3Wi W2 + aia2W3 + aia3W2 + a2 a3Wi = = (wi — ai)(W2 — a2 )(W3 — a3) + aia2a3 = 0,

f2 = W1W2 W3 — biW2W3 — b2WiW3 — b3WiW2 + bib2W3 + bi b3W2 + b2 b3Wi = = (wi — bi)(w2 — b2)(w3 — b3) + bib2b3 = 0,

f3 = WiW2 W3 — CiW2 W3 — C2Wi W3 — C3Wi W2 + CiC2W3 + CiC3W2 + C2C3Wi = = (wi — Ci)(w2 — C2)(W3 — C3) + Ci C2C3 = 0.

The Jacobian of the system (30)

A = (W2 — a2)(w3 —a3)[(wi — bi)(w3 — b3)(wi —Ci)(w2 —C2) — (wi —bi )(w2 — b2)(wi —Ci)(w3 — C3)] —

— (wi — ai)(w3 — a3)[(w2 — h)(w3 — b3)(wi — Ci)(w2 — C2) — (wi — bi)(w2 — b2 )(w2 — C2)(w3 — C3)] +

+(wi — ai)(w2 — a2 )[(w2 — b2)(w3 — b3)(wi — Ci)(w3 — C3) — (wi — bi)(w3 — b3 )(w2 — C2)(w3 — C3)].

Then, by Theorem 5, we obtain J(0,0,0) = Y1 j( — 1)s(J)

( — 1)\\k\\ r WiW2W3 ■ (aia2a3)kl (bib2b3)k2 (CiC2C3)k3 ■ A

(2m)2 J (wi — ai)kl+i(w2 — a2)kl+i(w3 — a3)kl+i X

\\k\\<2 p

r q,aj

(30)

dwi A dw2 A dws

(wi - bi)k?+i(w2 - b2)k?+i(w3 - b3)k2+1 ■ (wi - ci)k3+i(w2 - c2)k3+1 (w3 - c3)k3+1'

(31)

where Fq,aj are cycles of the form {|wi - ai\ = rn, \w2 - b2\ = r22, w - c3\ = r33}; {|w3 - a3\ = ri3, \wi - bi\ = r2i, \w2 - C2I = r32}; {\w2 - a2\ = ru, \w3 - b3\ = r23, \wi - ci\ = r3i}, taken with a positive orientation, and {^ - a^ = rii, \w3 - b3\ = r23, \w2 - c2 \ = r32};

{\w2 - a2 \ = ri2, \wi - bi\ = r'2i, \w3 - C3\ = r33}; {^3 - a3\ = ri3, \w2 - b2\ = r22, \wi - ci\ = r3i}

with negative orientation.

Calculating these integrals, we get

-a(i,i,i) = J(0,0,0) = ai b2c3 + aib3c2 + a2bic3 + a2b3ci + a3bc + a3b2 ci+ (32)

+

+

a3CiC2C3 a3 - C3

a2bib2b3

a2 - b2

bi

+

b2

bi - Ci b2 - C2

C3

+

Ci

C3 - b3 Ci - bi

+

+

aibi b2b3

ai - bi

a3bib2b3 a3 - b3

C3

+

C2

C3 - b3 C2 - b

C2

+

Ci

C2 - b2 Ci - bi

+ ■

ai Ci

ai - Ci

b2C2C3 + b3C2C3 + a2a3 b2 + a2a3b3

+ ■

a2C2 a2 - C2

b2 - C2 b3 - C3 a2 - b2 a3 - b3

biCiC3 + b3CiC3 + aia3b3 + aia3bi

+

bi - Ci b3 - C3 a3 - b3 ai - bi

+

+

So, we found the sums of the roots ^(i<iti) without calculating the roots of the system themselves.

3. General systems of transcendental equations

Let fi(z),..., fn(z) be a system of functions holomorphic in a neighborhood of the origin in the multidimensional complex space Cn, z = (zi,..., zn) .

We expand the functions fi(z),..., fn(z) in Taylor series in the vicinity of the origin and consider a system of equations of the form

fj (z) = Pj (z) + Qj (z)=0, i = l,...,n, (33)

where Pj is the lowest homogeneous part of the Taylor expansion of the function fj (z). The degree of all monomials (with respect to the totality of variables) included in Pj, is equal to mj, j = 1,... ,n. In the functions Qjl, the degrees of all monomials are strictly greater than mj.

The expansion of the functions Qj, Pj, j = 1,... ,n in a neighborhood of zero in Taylor series converging absolutely and uniformly in this neighborhood has the form

Qj (z)= E alza, (34)

\\a\ \ >mj

Pj (z)= E b1 z13, (35)

\\fi\\=mj

j = 1,...,n

where a = (ai,... ,an), ¡3 = (pi,..., f3n) are multi-indexes, i.e. a.j and ¡3j are non-negative integers, j = 1,... ,n, ||a|| = ai + ... + an, \\fl|| = fi + ... + fn, and monomials za = z^1 ■ z■■■ zan, z3 = zf1 ■ z^2 ■■■ z3n.

In what follows, we will assume that the system of polynomials Pi(z),..., Pn(z) is nonde-generate, that is, its common zero is only point 0, the origin.

Consider an open set (a special analytic polyhedron) of the form

Dp(r1,...,rn) = {z : \Pj(z)\ <rj, i = j,...,n}, where r1,... ,rn are positive numbers. Its skeleton has the form

rp(ri,...,rn) = rp(r) = {z : \Pj(z)\ = rj, j = 1,...,n}. Let us start with a statement. Lemma 3. The next equality is true

T = 1 [_1_ fi df2 dfn =

T = (2ni)n J zj1 + 1 • z]2+1 ••• zYn+1 ^ fi A f2 A ... A fn

Tp

= f WJ1 + 1 • WY22 + 1 • • • wZn+1 • f A f A ... A f = (-1)n J .

(2ni)n J 1 2 n f 1 f2 fn '

r p

For what follows, we need a generalized formula for transforming the Grothendieck residue.

Theorem 6. Let h(w) be a holomorphic function, and the polynomials fk (w) and gj (w), j,k = 1,... ,n, are related by the relations

9j

Eajkfk, j = 1,

k = 1

the matrix A = \\ajk ||,nk=1 consists of polynomials. Consider the cycles

rf = {w : \fj(w)\ = rj,j = 1,...,n}, r9 = {w : \gj (z)\ = rj,j = 1,...,n},

where all rj > 0. Then the equality

n k

det A assj dw

J h(w) fw = £ J h(w)-sJ=-, (36)

K. t k„ =f. n<kM)!

s = 1 S,j = 1

holds. Here 0! = 01!02!... 0n, 0 = (01,02,... ,0n), the summation in the formula is over all

n

non-negative integer matrices K = \\ksj\\n j=1 with the conditions that the sum Y1 ksj = aj, then

s=1

n

0j = £ kjs. Herefa = fa • • • fa, gf = gf1 ••• gfc.

j=1

Theorem 7. The next formulas are valid

p 1

E

zYi + 1 zY2 + 1 zYn+1 j=1 Zj1 ^ Zj2 ••• zjn

(2ni)n) J wj1 + i • wT+i • • • wn~+1 •^AA^AA ... A

11 + 1 • w^2 + 1 ••• w7n + 1 • f A f A ... A f

f 1 f 2 fn

E

II a\\ ^ \\y\\ +n

n+ \

(2ni)

n1 r e

„Y1 + 1 „ „T2 + 1

,Yin + 1 -

A • Q?1 ■ Qa ■■■ Q ^ dw1 A dw2 A ... A dwn

P\

ai + 1 # p«2 + 1

... P°

P n

nn

EkSj I!

E ■

HKH^HYH+n

s = 1 \j=1

-M

n (ksj)!

sj = 1

vY+I ■ A ■ det A ■ Qa n

s,j = 1

w

j=1

ßj Nj + ßj +Nj

where \\K|| = Y1 ksj, and the functional M assigns its free term to the Laurent polynomial.

s,j = i

In fact, in Theorem 7, analogs of the classical Waring formulas for finding power sums of roots of a system of algebraic equations are obtained.

Consider a system of equations in two complex variables

2 = o,

\ /1(z1,z2) = a1Z1 — a2z2 + z

\/2 (Z1,Z2) = &1Z1 + b2Z2 + z| = 0.

(37)

It satisfies the conditions on Qj(z) We will assume that a1b2 + a2b1 = 0, i.e. the system of lower homogeneous polynomials is nondegenerate.

Let us change variables z1 = —, z2 = —. Our system will take the form

W\ W2

{/i = -a2w2 + a1w1w2 + w2 =0, f2 = b2wiw2 + biw2 + wi =0.

This system has 4 roots, on the coordinate planes there is one root — (0,0). The Jacobian of the system (38)

(38)

A

—2a2W1 + a1W2 a1W1 + 1 b2 W2 + 1 2b1W2 + b2W1

Notice that

—2a2 b2W2 — 4a2 b1W1W2 + 2a1b1W\ — a1W1 — b2W2 — 1.

Q1 = W2, Q2 = W1.

P1 = —a2 W2 + a1W1W2, P2 = b2 W1W2 + b1W^.

(39)

To find the matrix A we use Example 8.3 from [4]. We introduce the matrix

Res =

(- —a2 a1 0 0

0 —a2 a1 0

0 b2 b1 0

V 0 0 b2 b1

2

n

x

k

a

sj

The determinant A of the matrix Res is A = a2b1(a2b1 + a1b2). Let us calculate some minors according to example 8.3 from [4]:

A

1 —

A

3 —

A

1—

A

3—

-a2 a1 0 b2 b1 0 0 b2 b1

a1 0 0

-a2 a1 0

0 b2 b1

0 - a2 a1

0 b2 b1

0 0 b2i

-a2 a1 0 0 - a2 a1 0 0 b2

— -a2b2i — a\b\b2, A■

2—

2

— a^1,

A4 — —

A

2—

a1 0 0

b2 b1 0

0 b2 b1

a1 0 0

—a2 a1 0

b2 b1 0

a2 a1 0

0 b2 b1

0 0 b2

— —a1b\,

— —a2b\,

— —a2>b2,

A

4 —

—a2 a1 0 0 —a2 a1 0 b2 b1

— a2b1 + 0,10^2.

Therefore, the elements aij of the matrix A are

a11 — aA (Äw + A2w2^ — A ((—a2b2 — a1b1b2)w1 — a^w) ,

a12 — A ( ÄÄ3w1 + A4w2

a!hw1 1 —cj,2b\w2 -Ä-, a21 — A(Ä1w1 + Ä2w2) — -A-,

a22 — Ä (Ä3w1 + Ä4w2) — Ä (—a2>b2w1 + + a1a2b2)w2) .

1 Ä

Then it is easy to check that

wf = anP1 + a12P2,

We calculate det A :

a21P1 + a22P2.

1

det A — a (a2b2w'2 — a2b1w1w2 — a^1w|) .

By Theorem 7

J(

(0,0)

E

||K||<2

( — 1)\\K\\ • (kn + k12)\ • (k21 + k¡2)! . kn\ • k12! • k21! • k22]-

X®

A • det A • Qk11+k21 • Qk12+k22 • akl1 • okl2 • ok21 • ok22

lu

'■12 • a21

22

3(kii+ki2)+1 3(k21+k22)+1 w1 • w2

We denote A = a2b1 + a1 b2. Cumbersome but simple calculations (using the definition of the functional M) give that

J(

1

(0,0)

2a1b2 + 6a2b2

b32

+

+

8a-\ b

1b2

4

A 02b1Ä 02b1Ä2 b1Ä2 02Ä2 A2

a1b2 3a2b1

a2b1

bL

a2A2 A2 A2 b1A2 '

This work was supported by the Russian Science Foundation, grant Complex analytic geometry and multidimensional deductions. Number: 20-11-20117.

0

0

3

w

2

3

a

1

References

[1] L.A.Aizenberg, On a formula of the gereralized multidimensional logarithmic residue and the solution of system of nonlinear equations, Sov. Math. Doc., 18(1977), 691-695.

[2] L.A.Aizenberg, A.P.Yuzhakov, Integral representations and residues in multidimensional complex analysis, Amer. Math. Monographs, AMS, Providence, 1983.

[3] A.K.Tsikh, Multidimensional residues and their applications. Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992.

[4] V.I.Bykov, A.M.Kytmanov, M.Z.Lazman, Elimination Methods in Polynomial Computer Algebra, Springer, New-York, 1998.

[5] L.A.Aizenberg, A.M.Kytmanov, Multidimensional analogues of Newton's formulas for systems of nonlinear algebraic equations and some of their applications, Siberian Mathematical Journal, 22(1981), 180-189.

[6] V.I.Bykov, Modeling of the critical phenomena in chemical kinetics, Komkniga, Moscow, 2006 (in Russian).

[7] V.I.Bykov, S.B.Tsybenova, Non-linear models of chemical kinetics, KRASAND, Moscow, 2011 (in Russian).

[8] A.M.Kytmanov, Z.E.Potapova Formulas for determining power sums of roots of systems of meromorphic functions, Izvestiya VUZ. Matematika, 49(2005), no. 8, 36-45 (in Russian).

[9] V.I.Bykov, A.M.Kytmanov, S.G.Myslivets, Power sums of nonlinear systems of equations, Dokl. Math., 76(2007), 641-644. DOI: 10.1134/S1064562407050018

10] A.M.Kytmanov, E.K.Myshkina, Russian Mathematics, 57(2013), no. 12, 31-43. DOI: 10.3103/S1066369X13120049

11] O.V.Khodos, On Some Systems of Non-algebraic Equations in Cn, Journal of Siberian Federal University. Mathematics &Physics, 7(2014), no. 4, 455-465.

12] A.M.Kytmanov, E.K.Myshkina, On calculation of power sums of roots for one class of systems of non-algebraic equations, Sib. Elektron. Mat. Rep., 12(2015), 190-209 (in Russian). DOI: 10.17377/semi.2015.12.016

13] A.A.Kytmanov, A.M.Kytmanov, E.K.Myshkina, Finding residue integrals for systems of non-algebraic equations in Cn, Journal of Symbolic Computation, 66(2015), 98-110. DOI: 10.1016/j.jsc.2014.01.007

14] A.V.Kytmanov, E.R.Myshkina, J. Math. Sciences, 213(2016), no. 6, 868-886. DOI: 10.1007/s10958-016-2748-7

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17] A.M.Kytmanov, Ya.M.Naprienko, One approach to finding the resultant of two entire functions, Complex variables and elliptic equations 62 (2) (2017) , 269-286.

[18] A.M.Kytmanov, O.V.Khodos, On one approach to determining the resultant of two entire functions, Russian Math., no. 4, 2018, 49-59.

[19] A.A.Kytmanov, A.M.Kytmanov, E.K.Myshkina, Residue Integrals and Waring's Formulas for a Class of Systems of Transcendental Equations in Cn, Journal of Complex variables and Elliptic Equations, 64(2018), no. 1. 93-111. DOI: 10.1080/17476933.2017.1419210

[20] A.M.Kytmanov, E.K.Myshkina, Residue integrals and Waring formulas for algebraic and transcendental systems of equations, Russian Math., 2019, no. 5, 40-55.

DOI: 10.26907/0021-3446-2019-5-40-55

[21] A.M.Kytmanov, Algebraic and transcendental systems of equations, SFU, Krasnoyarsk, 2019.

[22] M.Passare, A.Tsikh, Residue integrals and their Mellin transforms, Can. J. Math., 47(1995), no. 5, 1037-1050.

О трансцендентных системах уравнений

Александр М. Кытманов Ольга В. Ходос

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Рассмотрены различные типы систем трансцендентных уравнений: простейшие, специальные и общие. Поскольку число корней таких систем, как правило, бесконечно, то необходимо изучить степенные суммы корней в отрицательной степени. Получены формулы для нахождения вычетных интегралов, их связь со степенными суммами корней в отрицательной степени, многомерные аналоги формул Варинга. Приведены различные примеры трансцендентных систем уравнений и вычислены суммы многомерных числовых рядов.

Ключевые слова: трансцендентные системы уравнений, степенные суммы корней, вычетные интегралы.