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

DOI: 10.17516/1997-1397-2022-15-2-137-149 УДК 517.55

Some Systems of Transcendental Equations

Alexander M. Kytmanov* Olga V. Khodosf

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.09.2021, received in revised form 22.10.2021, accepted 20.01.2022

Abstract. Several examples of transcendental systems of equations are considered. 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. Calculations of multidimensional numerical series are given.

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

Citation: A.M. Kytmanov, O.V. Khodos, Some Systems of Transcendental Equations, J. Sib. Fed. Univ. Math. Phys., 2022, 15(2), 137-149. DOI: 10.17516/1997-1397-2022-15-2-137-149.

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 in a negative power are considered for various 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.

* [email protected] https://orcid.org/0000-0002-7394-1480 [email protected] © 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 some systems of transcendental 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. General systems of transcendental equations

In this section we follow the paper [22].

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 a neighborhood of the origin and consider a system of equations of the form

fj (z) = Pj (z) + Qj (z)=0,

1,

(1)

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 that converges absolutely and uniformly in this neighborhood has the form

Qj (z)= E

\\a\\ >mj

Pj (z)= E j

(2)

(3)

\\ß\\=mj

j

i,

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

a2 _ . zan = Jil , _ . z2 zn , z = zi z2 zn .

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 the point 0, the origin.

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

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

rp(ri,...,rn) = rp(r) = {z : \Pj(z) \ = rj, j = l,...,n}.

Let us start with a statement. Lemma 1. The next equality is true

1 r 1

(2ni)n

Y1 +1 „12 + 1

' zn

f A f A f1 f2

A

dfn

(-l)n (2ni)n

„71 + 1 „,,72 + 1

f a f A ... A f = (-l)n.Lf.

f 1 f 2 fn

ß

n

1

2

1

2

n

For what follows, we need a generalized formula for transforming the Grothendieck residue (see [23]).

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

n

gj = E ajkfk, j = 1, 2,...,n,

k = l

the matrix A = \\ajk ||nk=-^ consists of polynomials. Consider the cycles

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

where all rj > 0. Then the equality

dw

h(w) fa =

ß!

n (ksj)!ra

s,j = 1

h(w)

det A as7 dw

s,j=1

9

ß

(4)

holds. Here /3! = 31!/2!... /3n, /3 = ■ ■ ■ ,/n), the summation in the formula is over all

n

non-negative integer matrices K = \\ksj\\nj=i with the conditions that the sum Y1 ksj = aj, then

3j = £ kjs. Herefa = ft1 ■■■ , g? = gf1 ■■■ gfr.

j=i

Theorem 2. The next formulas are valid

p 1

71 + 1 72 + 1 7^+1

j=1 Zj1 • Zj2

Zjn

(2ni)n) f WJ1 + 1 • w2>2+1 • • • wnnn+1 • f A f A ... A f J f1 f 2 fn

r P

(_1)n+IIa

E

(2ni)n J

Il a II ^ \\j\\ +n p _

wY 1+1 • wY2+1 ••• wnr+1x

A • Qa • Q%2 ••• Qann dw1 A dw2 A ... A dwn

p«l + 1 pa 2 + 1 pa„ + 1 P 1 • P 2 • • • Pn

nn

(_1)II-II+n n I Eksj ) !

E ■

II-IKIMI+n

s = 1 \ j = 1

-m

n (ksj)!

s,j=1

vY+¡ • A • det A • Qa n

s,j=1

n wßj

j=1

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

s,j=l

In fact, in Theorem 2, analogs of the classical Waring formulas for finding power sums of roots of a system of algebraic equations are obtained.

s = 1

s

x

2. Examples

Example 1. Consider a system of equations in two complex variables

Jfi(zi,z2) = ziz2 + bi zi + 62 z2 = 0, If2(zi,z2) = 1 + aizi + a2z2 = 0.

Let us replace the variables zi = —, z2 = —. Our system will take the form

Wi W2

ifi = 1 + b2Wi + biW2 = 0,

f2 = WiW2 + a2Wi + aiW2 = 0.

(5)

(6)

The Jacobian of the system (6) A is equal to

A

b2 b1, W2 + a2 W1 + a,1

b2W1 - 61W2 + (a\b2 - a2bi).

Note that

Let us calculate det A Since

Q1 =

Q2 = a1W2 + a2 W1.

) P1 = 61W2 + &2W1, I P2 = W1W2.

w1 = anP1 + a12 P2, w2 = a21 P1 + a22 P2,

where Pi = biW2 + b2Wi, P2 = WiW2.

Therefore, the elements of an are equal

Therefore,

By Theorem 2

W1 61

a11 = -¡—, a12 = —T~ ' 62 62

_ W2 62

a21 = ~¡—, a22 = —T~. 61 61

W2 W1 W2 61 - W1 62

det A = — - — =-—-.

62 61 61 62

(7)

(8)

J(0,0) = E

\\K\\=kii + ki2 +fc21+fc22^2

(-1)\\K\\ ■ (k11 + k12 )! ■ (k21 + k22)! . ku\ ■ k12! ■ k21! ■ k22 !

xM

A ■ det A ■ (Q\11+k21 ■ Ql12+k22 ■ ak11 ■ a^2 ■ a^f ■ aIf

2( kn + k12) 2(k21 + k22) W1 ■ W2

J(

(0,0) = E

\\K\ = kn + k12 + k21 + k22^2

(-1)\\K\\ ■ (kn + k12)! ■ (k21 + k22)! . kU! ■ k12! ■ k21! ■ k22 !

m

(_1)k12+k22 (w2&1 _ w162) • (b2w1 _ b1w2 + a1&2 _ a2&1)

T,1 + k21+k22-k12 l1 + fc11 + fc12-fc22 b1 • b2

(a1w2 + a2w1)fc12+fc22 fc11+2fc12 „,,fc21+2fc22

1

w

2

Calculate the value of the sums (0, 0, 0,0) :

(w2b1 _ w1b2) • (b2w1 _ 61 w2 + a162 _ a2b1)

m

b1 b2

(1, 0, 0, 0) :

m

_(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1)

b1b2 • w1

a1b2 _ a2 b1 b1b2 :

(0, 1, 0, 0) :

m

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1) • (a1w2 + a2w1)

b2 •w2

_a2(a1b2 _ a2b1) a2b1 --- = _a1a2 +

b2

(0, 0, 1, 0) :

m

_(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1 b2 _ a2^)

b1b2 • w2

a1b2 _ a2b1 b1b2 '

(0, 0, 0,1) :

m

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2^) • (a1w2 + a2w1)

b\ • w2

a1(a1b2 _ a2b1) = b =

_a1a2 +

a1b2

"bT '

(2 0 0 0) :

m

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1 )

1

b1b2 '

b1b2 • w1 (0, 2, 0, 0) :

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1 ) • (aw + ¿2w1)2 • b1

m

(0, 0, 2, 0) :

2 • w 1

a2b1

"bT '

m

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1 )

1

b1b2 '

b1b2 • w2 (0, 0, 0, 2) :

(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1 ) • (a1w2 + a2w1)2 • b2

m

(1, 1, 0, 0) :

b1 • w4

m

_2(w2b1 _ w1b2) • (b2w1 _ b1w2 + a1b2 _ a2b1 ) • (a1w2 + a2w1)

bf • wf

a1b2

"bT '

2a2 b2 '

0

2

(1,0,1,0) :

M

(W261 - W162) ■ (62W1 - 61W2 + a\62 - a261)

62 62 ■ W1W2

61 62

(1,0,0,1) :

M

-(W261 - W162) ■ (62W1 - 61W2 + 0,162 - a261) ■ (a1W2 + a2W1)

6162 ■

2

162 ■ W1W2

a2 2a1

(0,1,1,0) :

M

-(W261 - W162) ■ (62W1 - 61W2 + a^2 - a261) ■ (a1W2 + a2W1)

6162 ■ W^W2

62 61

a1 2a2

61

62

(0,1,0,1) :

M

(W261 - W162) ■ (62W1 - 61W2 + a162 - a261) ■ (a1W2 + a2W1)2

6162 ■ W2W2

a1262 a2261 61 62

+ 2a1a2,

(0,0,1,1) :

M

Therefore

-2(w261 - W162) ■ (62W1 - 61W2 + a162 - a261) ■ (a1W2 + a2W1)

^ 2

2a1

"67.

a2 a1 a2162 a2261

J(0,0) = 2a1a2 ^--^--

62 61 61

62

Example 2. Consider a system of equations in two complex variables

(fi = 1 + biW2 + b2Wi = 0, f2 = WiW2 + aiW2 + a2Wi = 0.

Let u = w1w2 , then Wi = —, substitute in our system

W2

b2 u

1 + bi W2 + — =0, W2 a2u

u + aiW2 +--= 0.

W2

multiply each equation of the system by W2

{W2 + biw2 + b2— = 0, W2 u + ai W2 + a2u = 0.

(9)

(10)

(11)

Now we multiply the first equation of the system by a1, and the second by b1 and subtract one from the other

W2(biu — ai) — u(aib2 — a2bi) = 0,

so

W2 =

u(a162 - a261)

biu — ai

Let us substitute this into the first equation of the system and get rid of the denominator and the second variable

62u(61u - a1)2 + u(a^2 - a261)(61u - a1) + 6\u2(a6 - a261)2 = 0.

2

We get

b\b2u2 + (bi(aib2 - a,2bi)2 + b1(a1b2 - a2bi) - 2aibib2)u + a\b2 - a1(a1b2 - a2b1) = 0. By the generalized Vieta theorem

^ (a162 - a,261)2 a1 a2 J(0,0) = 1^ Wj1 ■ Wj2 =--r-r-+ -T- + -T-.

j=1

162

Example 3. Consider a system (13) of equations in two complex variables (example 1). Recall the well-known expansions of the sine into an infinite product and a power series:

1

k=1

k2n2

E

k=0

kk

(-1)k z (2k + 1)!'

which uniformly and absolutely converge on the complex plane and have the order of growth 1/2.

Consider the system of equations

f1(z1,z2) = z1z2 + 61z1 + 62 z2 = 0,

, , sin^/az + a2z2 ~ / a1z1 + a2z2 .

f2 (z1, z2) = -, , - = H 1--—- I =°.

Va1z1 + a2z2 s = 1

(12)

Using the formula obtained above and the known expansion of the series, we obtain that the integral J0,0 is equal to the sum of the series

J(

(0,0)

Ea262 a261 + V^ a2 + ^^ a1

a2261

a2

a1

2a1a2

n4s461 ^ n4s462

s=1 1 s = 1 2

n2s262 n2s261

s=1 2 s=1 1 s=1

(a162 - a261)2 + a^ + a261

90b1b2 6b1b2

Example 4. Consider a system of equations in two complex variables

j f1(z1, z2) = a1z1 - a2z2 + z'2z2 = 0,

\f2(z1, z2 ) = 61 z1 + 62 z2 + z1z\ = 0.

(13)

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

Wi W2

f1 = -a2W2 + a1W1W2 + 1 = 0, f2 = 62W1W2 + 61W2 + 1 = 0.

(14)

The Jacobian of the system (6) A is equal to

A

-2a2W1 + a1W2 62 W2

a1W1 261W2 + 62W1

-2a262W2 - 4a261 W1W2 + 2a161W~2.

Note that

Q1 = 1,

Q2 = 1-

z

n4s4

Pi = —+ a\W\W2 = 0, P2 = &2W1W2 + bi w2 = 0.

(16)

Calculate det A :

Res =

—a2 ai 0 0

0 —a2 ai 0

0 b2 bi 0

0 0 b2 bi

A = a2bi (a2bi + ab)/. Let us determine the minors:

A1

A 3

Ai = —

A3 = —

— a2 ai 0

b2 bi 0

0 b2 bi

ai 0 0

— a2 ai 0

0 b2 bi

0 —a2 ai

— 0 b2 bi

0 0 b2

— a2 ai

— 0 a2

0 0

—a2b1 — aibib2,

a\b1,

A 2 = -

A 4

A

2=

= —a2 b2,

A

4 =

ai 0 0 b2 bi 0 0 b2 bi ai 0 0

—a2 ai 0 b2 bi 0

a2 ai 0 0 b2 bi 0 0 b2

—aibi,

= — a 2 b 22 ,

a2 ai 0 —a2 0 b2

ai bi

= a2bi + aia2b2.

Therefore, the elements of an are equal

1

aii

i = A (AiWi + A2W2 ) = A ((—a2bi — aibib2)wi — aibiw2)

A

ai2 = a1 (A3Wi + A4W^ = aib^i,

1 — a 2 b 22 w 2 a2i = A (AiWi + A2W2) = -A-,

a22 = A1 (A3Wi + A4W2 ) = -1 (—a'2b2Wi + (a2bi + aia^)^)

W3 = aiiPi + ai2 P2,

w\ = a2iPi + a22 P2,

were Ai = —a2W2 + aiwiw2, P2 = b2WiW2 + biw|. It is not difficult to make sure that

Then

A(1, 3) = — A(1,4) A(2, 4) A(2, 3)

a2 ai 0 b2

— a2 ai

b 2 bi

ai 0

b2 bi

ai 0 0 b2

= a2 b2, = —a2bi — aib2, = —aibi, = aib2.

0

0

0

0

Let us calculate now det A :

1

det A = a (A(1, 3)w2 + A(2,4)w2 + (A(2, 3) + A(1,4))ww) =

= i ($2^2w2 — a2b1w1w2 — aibiw?i) .

By Theorem 2

J

(0,0) = E

||K|| = fcii+fci2 + fc21+fc22^2

( —1)||K|1 • (fell + M! • (fe21 + k22)!

kn! • ki2! • foi! • ^22!

xM

A • det A • ak1\i • a^2 • a$fi • a^2

3(fcii+fci2)+1 ,„3(fc21+fc22)+1

We calculate the value of the sums by denoting A = a2b1 + a1b2, (0, 0, 0,0) :

"A • det A

M

w1 • w2

0,

(1, 0, 0,0) :

M

a11 • A • det A w4 • w2

—2a1 a2b2 b"2 — 2a2b1b2(a2b2 + a1b1b2) 2a1b2 2b2

A2

A2 A

(0,1, 0,0) :

M

a12 • A • det A w4 • w2

a2b1(2a2b1b2 — 4a2b1 b2) 2a1b2

A2

A 2

(0, 0,1,0) :

M

a21 • A • det A

4

w1 • w24

a2b2(4a1a2b1 — 2a1a2 b1) 2a1b2

A2

A2

(0, 0, 0,1) :

M

a22 • A • det A

4

w1 • w4

2a2ia22b21b 2 + 2a1a2b1(a2 b1 + a^b2) 2a1b2 2a1

A2

A2 A

(2, 0, 0,0) :

(0, 2, 0,0) :

(0, 0, 2,0) :

M

M

M

a21 • A • det A

x

1

2

0

0

0

(0,0,0, 2) :

(1,1,0,0) :

(1,0,1,0) :

(0,1,1,0) :

(0,1,0,1) :

(0,0,1,1) :

(1,0,0,1) :

Therefore

M

2M

M

M

M

2M

M

a22 ■ A ■ det A

7

W1 ■ W7

a11a,12 ■ A ■ det A

W7 ■ W2

a12a22 ■ A ■ det A

4 4 W4 ■ w4

a21a22 ■ A ■ det A

7

W1 ■ W2

o>110,22 ■ A ■ det A

J1 ■ w 2

0.

_ 2a162 2a262 262 2a1 2a\62 2a1622

J(0,0) = ^^ a^ - X - X + + ^

= 2a162(a1 + 62) 2(a1 + 62) + 2a162 (a1 + 62)

A2

A

A2

J(

(0,0)

2(a1 + 62)

A .

Example 5. Consider a system of equations in two complex variables

(17)

f1 = -a2W2 + a1W1W2 + 1=0, J2 = 62W1W2 + 6W + 1 = 0.

(18)

Let u = w1w2, then w2 = —, substitute in our system

Wi

-a2w'2 + a^u +1 = 0,

62u + 61 + 1 = 0.

(19)

We get rid of the denominators

-a2w'2 + a^u +1 = 0,

62uw2 + 61u2 + w2 = 0

0

0

0

0

0

0

Calculate det A :

A:

—a2 0 a1 u + 1 0

0 —a2 0 a1u + 1

b2 u + 1 0 b1 u2 0

0 b2U + 1 0 b1u2

A = u4(aj b2 + 2a1a2b1b2 + a2b2)+ +u3(2a1a2b1 + 2a2b1 b2 + 2a1b2 + 2a1 b¡) + u2(2a2b1 + af + 4a1b2 + b22) + u(2a1 + 2b2) + 1.

By the generalized Vieta theorem

J(

(0,0)

E<

j=1

Yl + 1 Y2 + 1

i 1 Oil 12 1

j1

j2

a3 2a1a2 b1 + 2a2b1b2 + 2afb2 + 2ab

a4

2

a2b1(a1 + b2) + a1b2(a1 + b2)

2

a2b2 + 2a1a2b1b2 + a2b2 (a1 + b2)(a1b2 + a2b1) 2(a1 + b2)

(21)

A2 A2 A

Example 6. Recall the well-known expansions of the sine into an infinite product and a power

series:

sin v 2

k=1 1 k=1

k2n2

E

k=0

(-1)

fc^fc

(2k + 1)1'

which uniformly and absolutely converge on any compact from the complex plane and have the order of growth 1/2.

Consider the system of equations

¡1(w1 ,W2) =

h(w1 ,w2)

sin \J-a1W1W2 + a2w'2

\J—a1W1W2 + a2W2 sin \Jb1W2 + b2W1 W2 ■\Jb1wl2 + b2W1W2

n 1 —

a1 W1 W2

+ a2W2\ =

k=1 \ k2n2 y

0,

.51 <1

b1W2 + b2W1W^ = 0

s2n2 J

(22)

Each of the functions of this system decomposes into an infinite product of functions from the system (18).

Therefore, the integral J0j0 is equal to the sum of the series

J(

(0,0)

E

k,s=1

2n2(a1 s2 — b2k2) a1b2 + a2b1

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

z

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Некоторые системы трансцендентных уравнений

Александр М. Кытманов Ольга В. Ходос

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Рассмотрены различные примеры систем трансцендентных уравнений. Так как 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. Вычислены суммы многомерных числовых рядов.

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