On some systems of non-algebraic equations in c n Текст научной статьи по специальности «Математика»

УДК 517.55

On Some Systems of Non-algebraic Equations in Cn

Olga V. Khodos*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 06.05.2014, received in revised form 09.06.2014, accepted 11.08.2014 A method of finding residue integrals for systems of non-algebraic equations containing entire functions is presented in the paper. Such integrals are connected with the power sums of roots of certain system of equations. The proposed approach can be used for developing methods for the elimination of unknowns from systems of non-algebraic equations. It is shown that obtained results can be used for investigation some model of chemical kinetics.

Keywords: non-algebraic systems of equations, residue integral, power sums.

Introduction

A method for the elimination n unknowns from a system of n non-linear algebraic equations (in the characteristic zero setting) based on multidimensional residue theory was proposed by L.Aizenberg [1]. Further developments of the method can be found in [2-4].

In general, the set of roots of a system of n non-algebraic equations in n variables is infinite. Moreover, multidimensional Newton series (with exponents in Nn) of the roots of such systems is usually divergent. In the paper, we connect residue integrals with specific systems of n non-linear equations and compute such residue integrals. Then we obtain from this computation (provided that such series do converge) the values of the sums of multidimensional Newton series (with exponents in (—N*)n) formed with the roots of such non-linear systems which do not belong to the union of coordinate planes.

A class of systems of equations containing entire or meromorphic functions was considered in [5].

The purpose of this paper is to generalize results given in [5] to a wider class of systems of non-algebraic equations; to obtain formulas for calculation of residue integrals and to reveal the connection between residue integrals and multidimensional power sums of roots.

1. Preliminaries

A.Kytmanov and Z.Potapova [5] considered the following system of functions:

fi(z), f2(z),...,fn(z),

where z = (zi, z2,... ,zn). Each function fj (z) is analytic in the neighborhood of 0 € Cn and has the form

fj(z) = z^' + Qj(z), j = 1, 2,..., n,

* [email protected] © Siberian Federal University. All rights reserved

where ßj = (ßj, ßj,... ,ßj) is a vector of integer nonnegative indices, zßj = zßl • zß2 • • • zßn, and \\ßj|| = ßj + ßj + ... + ßn = kj, j = 1,2,... ,n. Functions Qj are expanded in a neighborhood of zero into an absolutely and uniformly converging Taylor series of the form

Qj (z)= E

3 za a

aaz

a H >kj

where a = (a1, a2,..., an), aj ^ 0, aj £ Z, and za = z ^1 • z^2 The formulas for calculation of residue integrals

_1 df

lj(r) '

Jß (2*i)nLr) Zß+U f

in terms of coefficients of Qj(z) were obtained.

Our goal is to obtain similar results in a more general case.

n

2. Calculation of residue integrals

We consider a system of functions f1 (z), f2(z),..., fn(z). They are analytic in a neighborhood of the point 0 £ Cn, z = (z 1, z2,..., zn) and has the form

fj (z) = (zf + Qj (z))epj =1, 2,...,n, (1)

where [j = ([j, [¡2,..., ) is a vector of integer nonnegative indices z32 = zf1 • z22 • • • zfn and \\[j|| = [j + [2 +... + [n = kj, j = 1,2,... ,n. Functions Qj, Pj are expanded in a neighborhood of zero into an absolutely and uniformly converging Taylor series of the form

Qj (z)= E <za, (2)

a3 za

II a II >kj

Pj(z) = E jzY, (3)

IIyII^0

where a = (a 1; a2,..., an), aj > 0, aj £ Z, and za = z f1 • z• • • z; Y = (Y1,Y2, ■ ■ ■, Yn), Yj ^ 0, Yj £ Z, and zY = zj1 • z]2 • • • z'^.

Firstly this system was considered in [6,7].

So the degree of all monomials in Qj greater then kj, j = 1,... ,n.

Consider the integration cycles Y(r) = Y(r1,r2,... ,rn), that are skeletons of the polydisks:

Y(r) = {z £ Cn : | zs | = rs,s = 1, 2,...,n}, r1 > 0,...,rn > 0.

For sufficiently small rj, cycles Y(r) lie in the domain where functions fj are analytic. Therefore, the series

E kk1 ••• ra

\ \ a \ \ >kj

E I j K • • rnn

lhll>o

converge for j = 1, 2,... ,n. Then, on the cycle y (tr) = Y(tr1 ,tr2,..., trn), t > 0, we have

|z|fj = tk3 • rf • rf2 ••• rfn = tk3 • rfj

and

IQj (z)| =

E <za

I a II j

< E t||a||K|ra < tkj+1 E |«a|ra,

|| a|| >fcj || a|| >fcj

0 < t < 1, j = 1,...,n. Therefore, for sufficiently small positive t, the following inequalities hold on the cycle Y(tr):

|zf > IQj(z)|, j = 1,2,...,n. (4)

Thus,

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

In what follows we assume that t = 1. Consider the system of equations

'fi(z) = 0, f2(z) = 0,

fn(z) = 0.

In general, system (5) can have non-discrete set of roots.

It follows from (4) that for sufficiently small rj the following integrals exist:

f 1 df f 1 df1 A df2 A dfn

(5)

J f J zf1 + 1 • zf2 + 1 ••• zn-+1 fi f2 fn'

where fy > 0, > 0,...,> 0, fy G Z, U = (1,1,..., 1). We call such integrals the residue integrals. These integrals are not the standard Grothendieck residues, since the cicle Y(r) does not connect with fuctions f 1,..., fn. The Logarithmic Residue Theorem is not applicable to such integrals as well.

These integrals do not depend on (r1,..., rn) under condition (4) on Y(r). Let us introduce the following notations

t= 1 [ f

P (2ni)n ,/7(r) ' f •

and fj(z) = z^ + Qj(z), j = 1,..., n.

Let us assume that Is is a vector of indices. The vector has n components and consists of s ones and n — s zeros (s = 0, ...,n). More exactly, each Is = I [i1,...,is] =

il is

(0,..., 0, 1, 0,..., 0, 1, 0,..., 0) G ({0,1})n where i1,...,is are the places of "one" in Is, 1 < i1 < ... < is < n. In what follows A/s stands for the Jacobian matrix of the system of functions such that to each "one" on the j-th place in Is there corresponds j-th row of the derivatives ^dfj/dz^, 1 ^ i ^ n in A/s and to each "zero" on the k-th place in I there corresponds k-th row of the derivatives (dPk/dzi), 1 < i < n in A/s.

Theorem 1 ( [6,7]). Under the assumptions made for the functions fj defined by (1), (2), (3) the following relations are valid:

J = E E E (—1),as" ••

s=0 Is ||as IKIßl+minCsjfcjj +... + kis )

(ß +(af + 1)ß^ + ... + (aS + 1)ß'S )!

d's (Ais • Qa (Is))

dzß+(«S + 1)ßi1 +■

or

Jß

EE

(aS + 1)ß

||as||M

Ai s • Qa (Is)

zß+(aS +1)ßl1 +... + (as + 1)ßi

(6)

(—1)

s=0 Is ||as ||<||,||+min(n,fcil +... + fcis )

where as is a vector of indices with s components; i| is the index of the k-th 1 in Is; 1s =

+ (a1 + 1)fyi1 + ... + (as + 1)fyis||; fy! = ft! • ft! • • • ft!; Qa(Is) = Q$ • Q§ • • • Q,f; d ||y||<^ dYl+...Yn ^

d Y = d Yl d Y2-d nn ; and M is a linear functional that assigns constant term to a Laurent

polynomial.

Remark 1. According to the proof the relation given in the statement of Theorem 1 contains only a finite number of coefficients of the functions Qj(z) and Pj(z).

Corollary 1 ( [7]). If all ft = (0, 0,..., 0), j = 1,..., n, then the integral J, is

Jß = EE E (-1)"

s = 0 Is ||as ||<||ß||

"M

EE E

s=0 Is ||as||^||ß

( —1)|asN d||ß| ß! äZß"

Ais Q(1s)a

Ai Q(1 )a

z=0

In the case of ft = (0,0,..., 0), it is also possible to obtain relation for J, with the use of the Cauchy integral formula for several complex variables, since fj (0) = 0 for all j = 1,..., n.

3. Power sums

Our next goal is to connect considered above integrals with power sums of roots of system (5). We must reduce the class of functions fj. At first we take Qj (j = 1, 2,..., n) as polynomials of the form

Qj(z) = E aaza, (7)

aEMi

where Mj is finite set of multi-indexes such that for a e Mj coordinates ak < ßj, k 1,2, ...,n, k = j, but ||a|| > kj for all a e Mj as before. Functions Pj (j = 1,2,.... are polynomials of the form

Pj (z)

E .

n)

(8)

0<||7||<pj

1

Let us introduce the substitution z7- = —, j = 1, 2,..., n. Therefore, we obtain

a( ^.... ^

\ W1 W2

j

1^11 1 -j + Qj — ,—,...,—

Wß V W1 W2 Wn

' y

1 fwS3 + Qj (w1, W2, .. ., w„)) ePj ( w 'w '

wßj +Sj \ j

where Sj is the degree of Wj, e1 = (1,0,..., 0), e2 = (0,1,..., 0), ..., en = (0, 0,..., 1), and degree of polynomials

(Qj(W1,W2,...,w„) = (Qj(w) = wßj+sjej • Qj ( —, —,..., —

V wi W2 w„

0

z

a

ß

z

1

w1 ' w2

e

w

n

is less than sj.

According to the Bezout theorem the system of nonlinear algebraic equations

fj (w) = wj + Qj (w)=0, j = 1, 2,...,n, (9)

has a finite number of roots that equals to s1 • s2^ • • sn and it has no roots on the infinite hyperplane CPn \ Cn.

Let us denote roots of system (5) not lying on coordinate planes as w(k) = (wT(k),

i-1 1

W2(k),■■■ ,wn(k)), k = 1, 2,...,M, M < si • s2^ • • Sn. Then points z/k)

\w1(k) w2(k)

are the roots of system (5), not lying on coordinate planes. So we have the fol-

wn(k) ,

lowing assertion

Lemma 1. System (5) with polynomials Qj of the form (7) and Pj of the form (8) has a finite number of roots z(1), z(2),..., z(M) not lying on coordinate planes {zs = 0}, s = 1, 2,...,n.

Let us introduce notation

M1

k=1 z1(k) 2(k) • • • zn(k)

This expression is the sum of roots of system (5) to negative powers. The roots are not lying on coordinate planes.

Theorem 2. For system (5) with polynomials Qj of the form (7) and Pj of the form (8), for which

l1 + ... + ln < [, (10)

where lj = (ljT,... ,lj) and lj is the degree of polynomial Pi with respect to variable zj; i,j = 1, . . . , n, the relation

J3 = (-1)n<7P+i,

holds (multi-index a ^ [, if this inequality is true for all coordinates).

Proof. We perform the substitution of variables zj = —, j = 1,2,... ,n in integral Jf. With

wj

this substitution the cycle Y(r) is transformed to the cycle

( 11 1

(-1)nY - - ..., - = (-1)ny(Rt,R2, ■ ■■,Rn)■

r1 r2 rn

Let us denote multi-index [j + Sjej as Yj, j = 1, 2, ... ,n. Then

dfj\Wl, w2,■■■, Win) dfj(w) .A j dwk A 1 / ,

-f-T^—T) = j)- £ -"■■w- £ wr(Pj>dw

j V W1 W21 ' WnjJ k = T k=T

Therefore

Jf = 7-^ ! wf+i( dfwl - E Yl • ddwk - E • (Pi)[Zh)dv>k) A

33 (2ni)n Jy(R) \fi(w) <k wk w2k '(Zk) 1

. / dfn(w) y^ n V^ 1 ir> V ,1

. I Xiwr—£14-ST—£ WT^-W^

We can easily show that all integrals of the form

' flj-rdfi(w) df/(w) dwj dwj ,

n,+ / _- A A ^_- A _j1 A A _

not containing

w,+/ ^ ; A ... A-^^- A^^ A ... A—^ (11)

f1(w) fj(w) Wjl Wj„-,

y(r)

n 1 ' EW2 •(P/)(zk)dwk

4=1 -

vanish if 0 < l < n and Rj are sufficiently large.

In a similar way we can prove that if integrand expression contains the differentials dPj and

—- then these integrals also vanish.

Wfc

Then we show that all integrals of the form

f + df1(w) dli(w) ( 11 1 \ ( 11 1 \ , N

w,+/ iu y A ... A y A dP/+W —,—,...,— A ... A dPn—,—,...,— (12) 77(R) f1(w) f;(w) \W1 W2 WnJ \W1 W2 WnJ

with condition (10) vanish if 0 < l < n and Rj are sufficiently large. Thus, we have

J, = (—ii w,+/dm A ... A fn(w)

(2ni)n 7 /i(w) /„(w)

y(r)

According to the Yuzhakov Theorem on Logarithmic Residue the last integral is equal the sum of values of holomorphic function w,+/ at all roots of system (9). However, the value of function w,+/ at the root of system (9), lying on coordinate plane, is equal to zero. Therefore, we obtain

J, = ( — 1)n^,+/.

Let us extend our consideration. Let us assume that functions fj have the form

fj1)(z)

f(2)(z)

/j (z) = , j = 1, 2,.... n, (13)

where fj(1) (z) and fj(2) (z) are entire functions in Cn of finite order of growth. They are represented by infinite product (uniformly converging in Cn)

if (z) = n ff)(z), fj2)(z) = n fjs2)(z).

s =1 s=1

Moreover, each factor has the form (z^s + Qjs (z))ePjs(z). Polynomials Qjs (z) and Pjs (z) are of the form (7), (8) and degrees of all polynomials degPjs ^ p, j = 1, 2,..., n, s = 1, 2,..., to. Thus fj(1)(

z) h fj2) (z) are entire functions with finite order of growth not greater than p. For all set of indexes j, ..., jn, where j, ..., jn G N, and each set of numbers i1,..., in, where i1,..., in are equal to 1 or 2, systems of non-linear algebraic equations

f(jl)(z) = 0, f2j2)(z) = 0, ..., fji)(z)=0, (14)

have (according to Lemma 1) finite number of roots not lying on coordinate planes.

Number of roots of such system is not more than countable set. Let us denote the roots as z(1), z{2),...,zw,....

Let us introduce the following expression

__

^1 + 1 rfc + 1 „Pn+1 ■

l = 1 z1(l) 2(l) • • n(l)

Here [31,...,[3n are nonnegative integer numbers and the sign of el is equal to +1 if the system of

(2)

the form (14), which root is z(l), contains even number of functions fjjj; and the sign of el is equal

to -1, if the system of the form (14), which root is z(l), contains odd number of functions f^.

For system (5), which consists of functions of the form (13), the points z(l) are roots or singular points (poles). All functions fj are analytic in some neighborhood of 0.

Let us introduce multi-undex lj = (lj ..., lj), where lj is the maximum degree of polynomial Pi with respect to variable zj; i,j = 1,... ,n contained in decomposition of fi (multi-index a < [ if this inequality valid fore all coordinates).

Theorem 3. Let us assume that the degrees of all polynomials Pj used in decomposition of functions of the form (13) in system (5) are bounded by number p and inequality

l1 + ... + ln <

holds. Then the following relations

Ji3 = (-1)n(Ti3+I

are valid.

The proof of this theorem immediately follows from Theorem 2.

4. Model of Zel'dovich-Semenov

We show that the considered methods of complex analysis can be useful in the study of the equations of chemical kinetics.

Consider the model of Zel'dovich-Semenov ideal mixing reactor (see. [9, Ch. 2, Eq. (2.2.1)]. It has the form

, . _y_

(1 - x)e (1+^y)--= —,

Da dr

(1 - x)e OW - = Y~T, Se dr

where [, D, a, S, e are positive parameters.

Denote Da = a, Se = b. Stationary states of the system satisfy the equations

' , s y x (1 - x)e (1+^y)--=0,

, ^ a (15)

(1 - x)e (1+^y) - - =0.

b

In [9, ra.2] qualitative study of the system conducted(15). We consider here a quantitative study.

From the Equations (15), we obtain that x = ^y. Substituting this expression into the first equation, we have

y y

e 1+0y

b — ay 461 -

We make the substitution

then y

zb

b + az

. Hence we have

We introduce the notation

b — ay b

z z

e 1 + z(ß+a/b) = _ .

b

1 a

b = Y ß + b = a

(16)

(17)

i.e. b = —, a = (a — @)b. Then from (17) we obtain the equation

Y

e i+az = Yz.

(18)

First examine the function

for positive z. Find the derivative

t \ 1 -z-

f(z) = — ■ e 1+az

ip'(z) = e 1+°

—a2 z2 — z(2a — 1) — 1 z2(1 + az)2

Investigate quadratic trinomial in the numerator of the fraction on the mark. Obtain that its discriminant D = 1 — 4a, then at 0 < a < 1/4 derivative p'(z) has two roots < z2, and at a > 1/4 is one root. Exploring the position of the vertex of the parabola, we obtain that for a < 1/2 it is positive, and for a > 1/2 it is negative.

Therefore, if the derivative has two roots, they are both positive. In this case, the smaller root z\ is a minimum point, and the larger root z2 is a maximum point.

Asymptotes of the functions p(z) are: z = 0 is vertical asymptote (p(z) ^ as z ^ +0), and the axis OZ is the horizontal asymptote (p(z) ^ +0 as z ^ +ro). Consider the equation

p(z) = Y, (19)

equivalent to Equation (17).

From the previous studies, we obtain that Equation (19) at 0 < a < 1/4 has three roots at p(z\) < y < p(z2). And if 0 < a < 1/4, Equation (19) has one root, when either z > p(z2), either z < p(z\).

At a > 1/4 Equation (19) has one root for all y, since the function p is strictly decreasing. Calculating z\ and z2 at a < 1/4, we obtain

zi

1 — 2a — a/D 2ä '

zi

1 — 2a + yfD 2ä '

D = 1- 4a.

Then

and

1-VD 2a2 ip(zi) = e 2a2 ■ -

1 — 2a — a/D'

i+VD 2a

tf(z2) = e 2 a2 ■ -

1 — 2a + VD'

y

z

2

Proposition 1. Let D = 1 — 4a > 0. Equation (19) has three positive roots at

1-Vd 2a2 2a2 e 2a2 •-— < y < e

1 — 2a — a/D 1 — 2a + a/D' one root if either

i+^p 2a2

Y > e 2a2 • -

1 — 2a + vD'

either

l-yp 2a2

Y < e 2a2 • -— .

1 — 2a — a/D

If D = 1 — 4a < 0, Equation (19) has only one positive root. Returning to the variables a, b, fy we obtain

Corollary 2. If D' = fy + a < 1/4, then Equation (17) has three positive roots at

b

-l+^l 1 — 2(fy + a/b) — VD7 , -l-y^; 1 — 2(fy + a/b) + VD7 2(fy + a/b)2 2(fy + a/b)2 '

has one positive root, if either

-i+*D2 1 — 2(fy + a/b) — VD7 , e 2(i+«/b)^-—LJ-*- < b

2(fy + a/b)2 <0'

either

, -l-^^l 1 — 2(fy + a/b) + a/D7

b < e 2(?+a/b)^ --7 7 -

2(fy + a/b)2 • At fy + "a > 1/4 Equation (17) has one positive root.

Thus, the system (15) has no more than three roots with positive coordinates. Let us consider how the system (15) has complex roots.

y t

Solving it by making the change t =-— (i.e. y =-—), we get

1 + py 1 — pt

t 1\ t

— e4 +

b(1 - ßt) a) ab(1 - ßt) t

Hence

b(1 - ßt)

(at - b(1 - ßt))e4 +1 = 0. (20)

Denote by

^(t) = (at — b(1 — fyt))e4 + t. Recall Hadamard theorem for functions of finite order of growth (see, for example, Definition 1. Expressions E(u, 0) = 1 — u,

2 p

E (u,p) = (1 — u)e"+"2r+-+V; p = 1, 2,... are called primary factors.

If the function f (z) in the complex plane has a finite order of growth, then there is not depending on n an integer p < p that the product

ft E (t'p) (21)

n=1 K '

converges for all values of z, if the series converges

p+1

E ^ <22)

r r,

where r1, r2,... are modules zeros of function f (z), and this series converges for all values of r, if p + 1 > p.

Definition 2. Product (21) with the least of the integers p for which the series converges is called the canonical product, constructed from the zeros of f (z), and is the smallest p is called its genus.

Theorem 4 (Hadamard). If a function f (z) is entire of order p with zeros z1,z2,..what is more f (0) = 0, then

f (z) = eQ(z)P (z), (23)

where P(z) is canonical product constructed from the zeros of f (z), and Q(z) is polynomial of degree not higher than p (see, for example, [8]).

Function ^(t) is a entire function of the first order and exponential type 1. Let function ^(t) have a finite number of zeros in C. Then by Hadamard's theorem it has the form

m = e ■ pn(t),

where a polynomial

«•M1 - £)■-O- i

and t1, t2, ..., tn are zeros of function ^(t). Then

(at - b(1 - pt))e* + t = e* ■ Pn(t).

Hence

t

at - b(1 - pt) - Pn(t)'

which is impossible since the right is a rational function.

Thus the number of zeros of ^(t) is infinite. These zeros have no limit points in C. If they are denoted by t1, ... ,tn,..., their modules |tn| — ro as n — ro.

Denote by (xn, yn) (n = 1,...) are roots of the system (15). Since y = --—, then yn — - —

1 - pt p

as n —> ro. Since ^(tn) = 0, then

etn

hence e*

atn - b(1 - ßtn) ' 1

a + bß'

n- t _t a

Smce x = 1 - Ki-ßT) e_'then bßß •

tn

Proposition 2. System (15) has an infinite number of complex roots (xn,yn) G C2, n = 1,.... there is a limit to this sequence of complex zeros when n ^ to and is equal to — ( —, — ) .

PJ

Let us consider the order of convergence of zeros yn. Since the function ^(t) is a first order, then (see, for example, [8]) |t < to for all e > 0.

n=1

Hence we obtain Corollary 3. Series

œ l 1+e

E

—+ß У"

< то for all e > О.

This work was supported by the Russian Foundation for Basic Research, Grants 12-01-00007.

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, Trans. Amer. Math. Monographs, AMS, Providence, RI, 1983.

[3] A.K.Tsikh, Multidimensional residues and their applications. Trans. Amer. Math. Monographs, AMS, Providence, RI, 1992.

[4] V.Bykov, A.Kytmanov, M.Lazman, M.Passare, (ed), Elimination Methods in Polynomial Computer Algebra, Math. and Appl., Kluwer Acad. Publ., Dordreht, Boston, London, 1998.

[5] 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).

[6] A.M.Kytmanov, E.K.Myshkina, Evaluation of power sums of roots for systems of non-algebraic equations in Cn, Russ. Math., 57(2013), no. 12, 31-43.

[7] 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.

[8] E.C.Titchmarsh, The Theory of Functions, Oxford Univerity Press, 1939.

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

О некоторых системах неалгебраических уравнений в Cn

Ольга В. Ходос

Рассмотрен метод нахождения вычетных интегралов для систем неалгебраических уравнений, состоящих из целых функций. Такие интегралы связаны со степенными суммами корней системы уравнений. Предложенный подход может быть использован для развития метода исключения неизвестных из систем неалгебраических уравнений. Показано, что полученные результаты могут быть использованы для исследования одной модели химической кинетики.

Ключевые слова: неалгебраические системы уравнений, вычетный интеграл, степенные суммы.

1