О НЕКОТОРЫХ ПРИМЕРАХ СИСТЕМ ТРАНСЦЕНДЕНТНЫХ УРАВНЕНИЙ Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2020-13-3-285-296 УДК 517.55

On Some Examples of Systems of Transcendent Equations

Alexander M. Kytmanov* Olga V. Khodosf

Siberian Federal University Krasnoyarsk, Russian Federation

Received 06.02.2020, received in revised form 16.03.2020, accepted 09.04.2020

Abstract. This article discusses examples of transcendent systems of equations of a general form. The residue integrals are determined over the cycles associated with the system. Formulas are given for their calculation and their relationship with the power sums of the roots of the system is established. Keywords: transcendent systems of equations, residue integrals, power sums of roots. Citation: A.M.Kytmanov, O.V.Khodos, On some Examples of Systems of transcendent Equations, J. Sib. Fed. Univ. Math. Phys., 2020, 13(3), 285-296. DOI: 10.17516/1997-1397-2020-13-3-285-296.

For systems of nonlinear algebraic equations in Cn, based on a multidimensional logarithmic residue, formulas were previously obtained for finding power sums of the roots of a system without calculating the roots themselves (see [1-3]). For different types of systems, such formulas have different forms. On this basis, a new method for the study of systems of algebraic equations in Cn is constructed. It arose in the work of L. A. Aizenberg [1], and its development is continued in monographs [2,4]. Its main idea is to find power sums of roots of systems (for positive powers) and then using 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 a formula (see [1]) obtained using the multidimensional logarithmic residue, to find the 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 transcendent 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, transcendent systems of equations arise, for example, in the problems of chemical kinetics [6,7]. Thus, the urgent task is to consider such systems.

In the works [8-15] power sums of roots are considered for a negative power for different systems of non-algebraic (transcendent) equations. To compute these power sums, a residue integral is used, the integration of which is carried out over skeletons of polycircles centered at zero. 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 is established.

Article [16] investigated more complex systems in which the lower homogeneous parts are decomposed into linear factors and integration cycles in residue integrals, and are constructed from these factors.

In [15], a system is studied that arises in the Zel'dovich-Semenov model (see [6,7]) in chemical kinetics.

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

The object of this study is transcendent 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 is established. See [16,17].

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

We expand functions f\(z),..., fn(z) into Taylor series in a neighborhood of the origin and consider a system of equations of the form

fi(z)= Pi(z)+ Qi(z)=0, i = !,...,-,

(1)

where Pi is the lower homogeneous part of the Taylor expansion of the function fi(z). The degree of all monomials (in the totality of variables) included in Pi, is mi, i = 1,... ,n. In functions Qi, the degrees of all monomials are strictly greater than mi.

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)

aj za

(2)

Pj (z)= E j

(3)

j

\\f\\=mj

1, . . . ,n,

where a = (ai,... ,an), 3 = (¡i,...,3n) are multi-indices i.e. a.j and ¡j are non-negative integers, j = l,...,n, ||a|| = ai + ... + an, ||3|| = ¡i + ... + ¡n, and monomials za = z^1 ■ z%2 ■■■ zan, z^ = zf1 ■ z%2 ■■■ .

In what follows, we will assume that the system of polynomials Pi(z),... ,Pn(z) it is non-degenerate, i.e. its common zero is only the point 0, the origin. Consider an open set (special analytic polyhedron) of the form

Dp (ri, ...,rn) = {z : \Pi(z)\ <ri, i = l,...,n},

where ri,... ,rn are positive numbers. Its skeleton has the form

rp(ri,...,rn)=rp(r) = {z : \Pi(z)\ = ri, i =l,...,n}.

These sets play an important role in the theory of multidimensional residues (see, for example, [2]).

For sufficiently small ri, the cycles rP lie in the domain of holomorphy of functions fi, therefore, the series

E \<\

l\rT

converge, i = 1,2,...,n. Then on the cycle rP (tr) = rP (tr i,tr2 ,...,trn) for sufficiently small t> 0, we have

and

\Pi(tr)

\Qi(tr)

E (tr)i:

E <(tr)c

\a\\ >mi

E

\\fi\\=mi

<

E

\\a\\ >mi

\\fi\\=mi

k r,

tWaW\ai0\ra = t

mi+ i

E

\\a\\ >mi

\ai0\rat\\

-(+1 )

all >m

a.

r

n

all >mi

m

1

t

f

m

Therefore, for sufficiently small t on the cycle Tp (tr) the inequalities hold

\Pi(z)\ > \Qi(z)l i =1, 2,.. .,n. (4)

Thus,

fi (z) = 0 Ha rp (tr), i =1, 2,...,n.

In what follows, we assume that t =1, that is, that the inequality (4) is valid on the cycle rp(ri,.. .,rn).

We introduce the concept of residue integral JY (see [18]). Define

_ _ df

(2*y—T)n J zY+r f rp

Jy = = (5)

1 df1 dfl A A dfn

(2nV—)nJ zj1 + 1 - zY2+1 ■■■ zln+1 fi f2 "' fn' rp

where y = (y1 ,.. .jn) is a multi-index. This residue integral is defined if r1,... ,rn is chosen so that the inequality (4) holds and the cycle rP does not intersect with the coordinate planes. Note that this integral is not a multidimensional logarithmic residue or a Grothendieck residue.

Recall some concepts from the space of the theory of functions C which equal to the product of n copies of Riemann spheres CP , i.e. C = CP1 CP1 .

Let zj : Wj be homogeneous coordinates in the j-th factor of the space C and let

Fj (z1,W1,...,zn,Wn)=0, j = 1,...,n (6)

be a system of equations consisting of polynomials Fj homogeneous for each pair of variables (zk ,wk), k = 1,... ,n. We will consider only those roots (z1, w1,... ,zn, wn) systems (6) for which

(zk,wk) e C2\{(0,0)}, k =1,...,n.

The roots of the system (6) with pairs having proportional coordinates determine one root (z1 : w1,...,zn : wn) in C . Let

a =(z(0) : w(0),...,zn°) : w^))

be the root of the system (6) for which all wf^ neq0. Then the point (z1,1,..., zn, 1) is the root of the system

Fj (z1,1,...,zn, 1) = 0, j = 1,..., n,

in Cn. Roots of a for which some wj0 are equal to zero correspond to infinitely remote roots in C .

For a given system of equations of the form (1) for which all fj (z) are polynomials, then in order to find the infinitely remote roots of this system in C , you must first go to homogeneous coordinates, substituting the zk/wk relationship instead of zk and discarding the resulting denominator, thereby obtaining a system of type (6). Solving it, we find ordinary roots and infinitely remote roots of the system (1).

We return to the consideration of the system (1). Assume that, in addition to non-degeneracy, the system P1(z),..., Pn(z) does not have infinite roots in the space C . We now consider as functions Qi(z), i = 1,... ,n, polynomials of the form

Qi (z)= E aLza. (7)

\\a\\ >mi

Suppose that for each i-th equation in (1) the conditions

degz, Pi < degz, Qi, Pi > Qi, j = i.

(8)

Here degzi P(z) is the degree of the polynomial P in the variable zi for the remaining variables We have deg Pi = mi. Denote deg Qi = si, a degz, Pi = mj, degz, Qi = sj. Then mi < si,

mi < si, i = 1,... ,n. In addition, mj > sj for j = i. Cases when ^ mj > mi.

j=i

1

In all functions, we write fi(z) = Pi(z) + Qi (z), i = 1,2,...,n, and replace zi = —, i

Wi

1,... ,n, assuming that all wi = 0. We get

Pif-,...,-)= E bf4r-^n

V Wi' 1 Wn ^ f wf1 wfn

\\f\\=mi i

Wn Wmi1 Wm

Eii m\-f i b f Wi ■■■

m- ^ f Wi

n \\f\=mi

-fn ,

and

11 Qi [—,...,—

W1 Wn

V" i —

a'a wai

Wnan

sn Wni

Ei s1 —ai aaW1

\\ a \\ >mi

si —an

■ Wni

We have

f,l-,-X)= P,(-L.....Q<( W-.....-L

,Wi -n J \—i WrJ V—i Wn

m„ xPi(W)+ Q i(W)

■ Wn i

(9)

where Pi are homogeneous polynomials

Pi(W i,..., Wn) = w'1 i ■ ■ ■ Wii ■ ■ ■ Wrnii ■ Pi ( —,..., — ) = i 1 n 1 i n i W1 Wn

s-—m- \ ^

bfWi

\ f\ =m

m\—f1 mn — fn

Wn

■Pi,

and Pi are homogeneous polynomials

Pi = £ b

,m1—f1

fW1

\ f\ =m

In PPi, neither w1, ..., nor Wn.

The polynomials <Qi have the form

Qi(wi,... ,Wn) = w^ ■ ■ ■ wsC

■ w'nH ■ Qi [ —,..., —

W Wn

m s m

W1 ■ ■ ■ Wi ■ ■ ■ Wn

-—_____L y

s1 sn Z^

i s —a1

aaWi

s- i

a \\ >m.

[wi\

mi si ■ Wn i i

E

i mi —a1 a awi i

mi an

■ Wn i

1

1

1

1

1

s

W

1

a >m

sm

=W

=W

mn—fn

. . W

n

sn—an

W

n

m1 — s1

W

1

a >mi

Denote by fi the functions

fi(w)= Pi(w)+ Qi(w)= wf< • Pi + Qi (w), i = 1, 2,...,n. (10)

We have

degPi > degQi, i =1,...,n. (11)

Consider a system of equations of the form (1) with polynomials Qi(z) satisfying the conditions (8).

Let Tp = Tp(e) denote the cycle

rp = {w e Cn : \Pi\ = ei, ei > 0, i = 1,...,n}. (12)

This cycle does not intersect with the coordinate planes for almost all ei, i = 1,... ,n. Consider the residue integral JY of the form

J = 1 i WY+I f (1/w)

J (2nV-l)nJ W f (1/w) '

where wY+T = wJ1 + 1 ••• w2n+1, f (1/w) = fi(1/wi,..., 1/wn) ••• fn(1/wu..., 1/wn), df (1/w) = = df1(1/w1,l. ., 1/wn) A ... A dfn(1/w1:..., 1/wn).

In fact, JY is obtained from the integral JY (5) using the substitution in the integrand zj = 1/wj, j = 1,...,n, and replacing rp by Tp. But the equality of these integrals needs to be proved.

Since the inequalities (11) hold for functions from the system (10), and the system of functions P1(w),..., Pn(w) is non-degenerate, the well known Bezout theorem says that the system of equations

fj (w)=0, j = 1,...,n, (13)

has a finite number of roots (counting each root so many times what its multiplicity is) and this number is equal to the product of the degrees of the polynomials Pj (w). We cite the theorem from [16].

Theorem 1. The following equality holds:

^ 1

zYi + 1 zY2 + 1 zYn+1 j=l Zjl ^ zj2 ••• zjn

E (-1)1

II a\\ ^ WyW +n

A ■ w Y1 + 1 ■ wY2+1. • • wnn+1 •

n pa.i + 1 pa.2 + 1 pan+1

P1 • P2 • • • Pn

dw,

where A is the Jacobian of the system (10).

For what follows, we need a generalized Grothendieck residue transformation formula (see [19], as well as [4, Ch. 2]).

Theorem 2 ([19]). Let h(w) be a holomorphic function, and the polynomials fk (w) and gj (w), j,k = 1,... ,n, be related by

n

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

k=1

the matrix A = \\ajk\\nk=1 consists of polynomials. Let us consider cycles

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

a

where all rj > 0.

Then the equality is valid:

dw

h(w) fa = E

0!

o )!r.

sj = 1

h(w)

det A aj dw

g

0

(14)

where 0! = 01!02!... 0n, 0 = (01,02, ■ ■ ■, 0n), the summation in the formula is over all integer

n

non-negative matrices K = \\ksj11n j=i with the conditions that the sum Y1 ksj = a.j, then 0j =

= S kjs. j=i

Here fa = f?1

70 = gO1 ■ ■■ g°n .

fan, gf = ai1

From this theorem, a statement is obtained in [16]. Theorem 3. The formulas are valid

i

-iy

,71 + 1 Y2 + 1

-,Yn + 1

(2nV-iy

,,Y1 + 1 „ „72 + 1

VYn n

dfn

+ 1 . f a f A ... A _ ¡1 ¡2 fi

£

(-i)

n+ \\a\\

\\ a\\ ^ \\Y \\ +n r ■

w11 + 1-w2l2+1'

A ■ Q?1 ■ Q?2 ■■■ dw1 A dw2 A ... A dwn

ÎDa1 + 1 Pa2 + 1 ÎDan+1

P1 ■ P2 ■ ■ ■ Pn

(-i)

\K\\+n

E ■

l|K\\<\\7\\+n

ksj

s=1 j=1

-M

(ksj

s,j=1

uY+r ■ A ■ det A ■ Qa f]

s,j=1

sj

n

j=1

w0j Nj +0j +Nj wj

where UK|| = Y1 ksj, and the functional M maps the Laurent polynomial to its free term. s,j=i

In fact, in Theorem 3, analogues of the classical Waring formulas for finding power sums of the roots of a system of algebraic equations are obtained.

Note that in [20] general algebraic systems of equations were considered, decompositions of their solutions in hypergeometric series were obtained. In addition, it proves analogues of Waring's formulas for systems of the form

ym + E XjV =0, Ai + ... + Xn < mj, j = 1,...,n,

\eA(j)u{0}

those higher homogeneous parts are monomials. We considered other (more general) systems of equations with functions of the form (10).

Consider a more general situation. Let the functions fj be meromorphic and have the form

fj (z) =

f(%) f(2)(z)'

j = 1, 2,...,n,

(15)

where f(1\z) and ff"\z) are entire functions in Cn that decompose into infinite products uniformly converging in Cn, fj2\o) = 0,

fr(z) = n tfM f (z) = n f%(z)

s = 1

s

1

2

jn

n

k

s

moreover, each of the factors has the form Pj,s(z) + Qj,s(z), and Qj,s(z) satisfy conditions (8), s = 1, 2,....

For each set of indices j 1,... ,jn, where j 1,... ,jn e N, and each set of numbers i 1, . . . , in, where i , . . . , in are equal 1 or 2, systems of nonlinear equations

fj(z) = 0, fj2(z) =0, ..., fj(z) = 0, (16)

have a finite number of roots not lying on coordinate planes.

The roots of all such systems (not lying on the coordinate planes) are no more than a countable set. Renumber them (taking into account multiplicities):

z(1), z(2),...,z(l),... .

Denote by ap+I the expression

aP+I = E Bi + 1 P + 1--pn+1 . (17)

l = 1 z1(l) • z2(l) ••• zn(l)

Here f31,... ,pn, as before, are non-negative integers, and the sign el is +1, if in a system of the form (16), the root which is z(l), includes an even number of functions fj2); and is equal to — 1 if in a system of the form (16), the root which is z(l), includes an odd number of functions

f (2)

For a system (16) composed of functions of the form (15), the points z(l) are roots or singular points (poles). All functions fj are holomorphic in a neighborhood of zero and are defined for them integrals Jp, since they have the form (1).

Theorem 4. For a system of equations with meromorphic functions (15) the series (17) absolutely converges, and

Jp = ( — 1)nvp+i.

Example 1.

Consider a system of equations in two complex variables

if1(z1,z2) = z1 — z2 + azf + bz3 = 0, f2(z 1,z2) = 1 + cz2 =0.

We make the change of variables z 1 = —, z2 = —. Our system will take the form

w 1 w2

\f1 = w 2w2 — w3 + aw 1 w2 + bw2 = 0, (18)

\J2 = w2 + c = 0.

The Jacobian of the system (18) A is AJ

2w 1 w2 — 3w 2 + aw2 w 2 + aw 1 + b 01

= 2w1w2 — 3w\ + aw2-

It is clear that _

{Q1 = aw 1 w2 + bw2,

Q2 = c.

I J1 = w2w2 — w3,

| P2 = w2.

Since

wwf = a,nJ1 + auP2, w2 = a21 P1 + a22PJ2, it is easy to show that the elements aij of the matrix A are equal

1 2

an = —1, a,12 = w1, a21 = 0, a22 = 1.

Thus, det A = —1. By Theorem 3

J(0,0) =

(-1)\\K\\ • (kn + kuy. • (k21 + k22lx

xffl

k-\-\ ! • k-2! • k2- ! • k22!

(3wl - 2wlw2 - aw2) • (awlw2 + bw2)kl1+k21 • ckl2+k22 • (-1)kl1 • (w2l)kl2 • 0k21 • 1k22

Simple calculations give that

w 3(kn+k12)+l w (k21 + k22 )-l W - • W 2

2

J(0,0) = c ■

Recall the well-known decomposition of the sine function into an infinite product:

00 / 2

sin z ^ _ z2

n 1 -

k2n2

k=1

which uniformly and absolutely converge on the complex plane and has a growth order of 1. Consider the system of equations

h(z1, z2) = z1 — z2 + az2 + bz3 = 0,

ft \ sin z2 n f2(z 1,z2) = - =0.

z2

Using the formula obtained above and the known sum, we obtain that the integral J(o,o) is equal to the sum of the series

^ 1 1

J(0'0) = 2^ ns? = 3.

s=1

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

\fi(zi, Z2) = z-Z2 + b-z- + b2Z2 = 0, [Î2(zi, Z2) = 1 + a-z- + a2Z2 = 0.

(19)

We make the change of variables z1 = —, z2 = —. Our system will take the form

w1 w2

I J = 1 + b2w1 + b1w2 = 0, 1 f2 = w1w2 + a2w1 + a1 w2 = 0.

The Jacobian of the system (24) Â is

Â

b2 b-L 2 + a2 L - + a-

= b2W- - b-W2 + (a-b2 - a2b-).

Notice that

Q1

Q2

i,

a1 w2 + a2w1.

We calculate det A : Since

I P1 = b1w2 + b2w1,

I P2 = w1w2.

w2 = auP1 + a12P2, wj = a21^1 + a22P2,

(21) (22)

where Pi = biw2 + b2wi, P2 = wiw2.

Therefore, the elements of aii are equal

aii = a2i

w1

w2 "b?

a12 =

a22 = -

b_l b2 '

b_l b1 .

Hence,

Notice that

w2 w1 wjb1 - w1b2

det A = 1T -TT =--.

b2 b1 b1 b2

Qi = —, Q2 =

Carrying out the same calculations as in the previous example, we obtain

J,

(0,0)

2(a1 + bj)

A '

Example 3.

Consider a system of equations in two complex variables

U1(z1, Z2) = a1Z1 - a2Z2 + zj =0, \f2(Z1, Z2) = b1 Z1 + b2Z2 + z\ =0.

(23)

It satisfies the conditions (8) on Qj(z). We assume that aib2 + a2bi = 0, i.e. the system of lower homogeneous polynomials is non-degenerate.

We make the change of variables zi = —, z2 = —. Our system will take the form

Wi W2

ifi = -a2w'2 + aiWiW2 + W2 =0, ¡2 = b2WiW2 + bi w\ + wi =0.

This system has 4 roots, on the coordinate planes there is one root, (0,0). The Jacobian A of the system (24) is equal to

(24)

A=

-2a2w1 + a1w2 a1w1 + 1

b2 w2 + 1

2bw + b2w1

-2a2b2wj - 4a2b1w1w2 + 2a1b1wj - a1w1 - b2w2 - 1.

Notice that

Q1 = w2, Q2 = w1.

(25)

P1 = —a2w2 + a\W\W2, p2 = &2W1W2 + b\w^.

(26)

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

Res :

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

(- -a2 a1 0 0

0 -a2 a1 0

0 b2 b1 0

V 0 0 b2 b1

- a2 a1 0

A1 = b 2 b1 0

0 b2 b1

a1 0 0

A 3 = - a2 a1 0

0 b2 b1

0 -a2 a1

A1 = 0 b2 b1

0 0 b2

= —a2 bi — aibib2,

A 2 = —

a1 0 0 b2 b1 0 0 b2 b1

= —a1b'\,

l!b1,

0, A

A 4 = -

a1 0 0

—a2 a1 0 b2 b1 0

2=

— a2 a1 0 0 b2 b1 0 0 b2

= — a 2 b 22 ,

-a2 a1 0 -a2 a1 0

A3 = - 0 -a2 a1 = —a2b2, A4 = 0 -a2 a1 = a2 b1 + a1a2b2

0 0 b2 0 b2 b1

Therefore, the elements a j of the matrix A are equal

an = A (A 1W1 + A2W2) = — ((—■a2b1 — a1b1b2)w1 — a^b^) ,

a 12 = a (A3W1 + A4W2

a1b1 W1

A

a22 = A (A3W1 + A4W2) = A (—■a|b2W1 + (a2b1 + a1a2b2^) .

1

A

a21 2

A (A1W1 + A2W2) = a2b2 W2

A

Then, it is easy to verify that

anP1 + a^P2,

a2p + a22P2.

We calculate det A :

det A = — (a2b2W2 — a2b1W1W2 — a1b1W^) .

By Theorem 3

J(

(0,0)

E

( —1)"K" • (k11 + k12)! • (k21 + k22)!.

xffl

l|K||<2

A • det A • Q1

kn! • k12! • k21! • k22!

kl1+k21 ^ Qfcl2 + k22 „kn „k

11

12

-,k21 21

k22 22

3(kii+ki2)+1 „„3(k21+k22)+1 W1 • W2

Denote A = a2b1 + a1b2. Cumbersome but simple calculations (using the definition of the functional M) give that

J(

(0,0)

1

A

2a1 b2 + 6a2b2

b32

+

+

8ai b'

1b2

4

a1b2 3a2b1 b2.

a2b1— a2b1—2 b1—2 a2-A2 A2 a2b1 a2A2 A2 A2 b1—2'

0

3

3

W

W

1

2

3

3

a

a

1

1

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О некоторых примерах систем трансцендентных уравнений

Александр М. Кытманов Ольга В. Ходос

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В данной статье рассматриваются примеры трансцендентных систем уравнений общего вида. Вычетные интегралы определяются по циклам, связанным с системой. Приведены формулы для их расчета, и установлена связь со степенными суммами корней системы.

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