Mellin transform for monomial functions of the solution to the general polynomial system Текст научной статьи по специальности «Математика»

УДК 517.55

Mellin Transform for Monomial Functions of the Solution to the General Polynomial System

Irina A. Antipova*

Institute of Space and Information Technology, Siberian Federal University, Kirensky, 26, Krasnoyarsk, 660074

Russia

Tatyana V. Zykova^

Institute of Space and Information Technology, Siberian Federal University, Kirensky, 26, Krasnoyarsk, 660074

Russia

Received 10.12.2012, received in revised form 10.01.2013, accepted 20.01.2013 In the present paper we give the calculation of Mellin transform for the monomial function of the vector-solution to the general polynomial system. We essentially use linearization of the system. In scalar case it defines bijective change of variables. In case of the system of equations we weaken requirements on the mapping given by the linearization: it is proper and its degree is equal to one.

Keywords: Mellin transform, algebraic equation.

Introduction

Consider a polynomial map

P = (P1,...,Pn) : Cn ^ Cn.

We assume that the sets A(j) c Z+ of exponents of the monomials in polynomials Pj are fixed while all the coefficients vary. Then we say that P is a general polynomial map from Cn to Cn. It defines a general system of polynomial equations of the form

E 4V = 0, i = i,...,n (1)

AeAW

with unknown y = (yi, ...,yn) G Cn and variable coefficients where C Z+ are fixed finite subsets, A = (Ai,..., An), yx = yJ1 .. .yJ".

In 1921 Mellin [1] presented an integral formula and series expansion for the positive power yM(x) of the function y(x) defined by the general (reduced) algebraic equation

ym + xiyx1 + ■■■ + xpyXP - 1 = 0, (2)

* antipova@akadem.ru t zykovatv@mail. ru © Siberian Federal University. All rights reserved

m > A1 > • • • > Ap > 1. This result was extensively applied by the Krasnoyarsk School (see, for example [2],[3]) in the investigation of the monodromy of the general algebraic function. The aim of our study is to calculate the Mellin transform for monomial function

1 1 M = (Mi, , Mi > 0, (3)

yf(-x) ' yf1 (-x) ••• y£n(-x) composed of coordinates of the solution y(x) to the reduced polynomial system of the form

yd(i) + ^ x^yA - 1 = 0, i = 1, .. ., n, (4)

AeAW

where the matrix of the columns of distinguished exponents =: D is non-

degenerate and A(i) := A(i)\ {d(i), 0} . As a rule, the system (1) can be reduced to the form (4) (see [4]). Now we consider the case of diagonal matrix D with positive integer diagonal elements m1,... ,mn. Let A be the disjunctive union of A(i), and let N be the cardinality of A, i.e., the number of coefficients in system (4). The set of these coefficients is a vector space CA = CA, where the coordinates of points x = (x^) are indexed by the elements A e A. We usually distinguish the group of coordinates corresponding to the indices A e A(i) by writing x(i). Let #A(i)

be the cardinality of the set A(i). We will assume that all sets A(i) lie in the interior of the simplex with vertices 0, m1e1,..., mnen, where e1,..., en is a basis in Zn. In other words all points A e A have nonzero coordinates and satisfy the following condition

n

]T(dW,A)< 1, (5)

i=1

where d(i) are columns of the matrix D := D-1.

In Section 1 we discuss the definition and the Jacobian of linearization (£, W) ^ (x, y) of system (4) which is the main tool in calculations of the Mellin transform for the monomial function (3). Section 2 contains the main results of the present paper. The first one is Theorem 1 which states that the mapping $ defined by the linearization (£, W) ^ (x, y) is proper and its degree deg $ is equal to one. The second one is Theorem 2 which gives the Mellin transform for the monomial function (3).

1. Linearization of the system (4)

Let Tn be the complex algebraic torus. We regard (4) as a system of equations in the space CA x Tn with coordinates x = ^x^j , y = (y1,...,yn) and introduce a change of variables (£, W) ^ (x, y) in CA x Tn by putting

xi° = -d°wA, A e A(i), i = 1,..., n, y = WD,

where W = (W1, ...,Wn), WA = ft W-(dW'A), WD = (w i(1),..., W, £ = . Therefore we can write (4) as a system of linear equations

Wi = 1+ E ^ i = 1,...,n. (7)

AeAW

(6)

If we define a change of variables £ ^ x in the space of coefficients CA by the formula

n I \ ^^

xiJ) = £$°n 1+ Z , A G A(i), i = 1,...,n, (8)

fc=i \ TeA(fc) / then vector y ( —x(£)) takes the following form

y (-*(0) = (w (e))D,

(9)

where W(£) = (W1 (e(1)) ,..., Wn (e(n))) is formed from the linear functions (7).

The idea of linearizing an algebraic equation by using a change of variables of this type is due to Mellin. It was realized by Mellin in [1] (see also [5]) in order to obtain the integral representation and series expansion for the solution of algebraic equation (2). In [6] this trick was applied to the "triangular" system when the first equation depends only on y1, the second one depends only on y1,y2 and so on. Analogous linearization of the system (4) was given in [7] with a view to calculate Mellin transform for the monomial function of the following type

yf(x) := yf (x) ■■■ (x), m = (^1,...,^n), Mi > 0.

The main result of [7] is the power series expansion of the function yM(x). This expansion was given using the formal calculation of the Mellin transform for M [yM(x)j.

Further we need to consider the restriction $ of the mapping C™ ^ C^ given by (8) on the positive orthant R^.

Lemma 1. The Jacobian of the mapping $ is equal to

d(x)

de)

n - E (d(i),A)-1

II W AeA(i) i=i

/

1 +

\

EE E

q=1 |1|=q teA1

1-

T.

1 -

(10)

where I is an odered set 1 ^ i1 < • • • < iq ^ n, t = (t41,..., Tiq ) is an element of Cartesian

product A1 = A(il) x • • • x A(iq), and £t = n £t* •

¿e/

Proof. The Jacobi matrix of the change of variables (8) has a block structure with n2 blocks. There are square blocks on the diagonal of the matrix. The diagonal elements of ith diagonal block are of the following type

dx

(i)

Ay =(l -(^.A^W-1) W-DA;

de:

the elements outside the diagonal in this block are

dx| = - (d(i),A) eA°w-1w-D A.

de

The nondiagonal (i, k)-block contains the following elements

dx

(i)

de

(k)

= - ^ d(i), ^ efw-1 w-D A, k = i.

T

T.

m

mu

e

T

q

T

m

A

The computation gives us

d(x)

M

nWi

i=i

- E. ,A>-1

det Wj - ]T

»(k) »A

AeA(fc)

j,k=1,n

where Sj is the Kronecker symbol, j, k = 1,..., n.

The last determinant may be reduced by the formula:

det(E + A) = 1 + ££

q=1 |1|=q

Ail ... Ail

Ail ... Ai" i1iq

where E is an unit matrix, A = (Aj) is an arbitrary matrix of order n, and I is an ordered set 1 ^ i1 < ... < iq ^ n.

All that now remains to be written is that

d (x) d(£)

n - E (d(i),A>-1

IIW AeA(i) i=i

i + EE

q=1 |1 | = q

Ai1 Ai1

Aiq Ai1

Aii1 iq

Aii

where Air = Y1 (S[ — \ d(ii), Ah £A , r, l = 1,..., q, and Sf is the Kronecker symbol. Hence we get the desired formula (10).

□

2. Mellin transform for the function

yM(-x)

We consider the monomial function (3) where ( — x) are branches with conditions yi (0) = 1, i = 1,..., n. Let us recall that Mellin transform of the function x) is defined by the following integral

M

1

(z)= /

JR

1

RN x)

y^(-x)

cz-i dx,

(11)

where x

z-I

,zi-1

-ZN-1

dx = dx1 • • • dxN (see, for example [5]). To calculate the integral

(11) we consider the transformation £ ^ x (or a mapping $ : R^ ^ R^) given by (8). Theorem 1. The mapping $ is proper. Its degree deg$ is correctly defined and deg$ = 1.

Proof. We prove that

N

) = dRN. Further we miss the upper index in the notations

xAi),£Ai), and write xA,£A for simplicity. Note that any coordinate plane £A = 0 is mapped to coordinate plane xA = 0. Moreover, the boundary points of the orthant are mapped only to the boundary points. If a sequence £(k) e R^, k e N, converges to the boundary point £ e dR^ then the sequence of images $(£(k)), k e N, also converges to a boundary point of the orthant. We are now going to prove that condition £(k) ^ implies x(k) = $(£(k)) ^ Note that coordinate xAk) may be finite in case when the corresponding coordinate £Ak) tends to but

(k)

:=E

(k)

AeA

y

1

1

+

A

when

e(k) = E €

(k)

AeA

Using assumption (5) for any A e A one can choose such a real positive n-dimensional vectors

A = (rj1), pA = (pA)

A__A |_A|

r

(rA), pA = (pA) that

^,A) + rA = pA, |PA I = 1-

Using the well-known Jensen inequality

«l1 - - < Pl«1 +-----+ Pn«n,

which is valid for any positive numbers a1; • • •, a„, p1; • • • ,pn, EPi get the following estimates:

(12)

1, and conditions (12) we

E4fc) = E €Ak) w(e(k))

AeA

AeA

-D A

E €Ak) (w (e(k))) ^ (w (e(k))) >

AeA

> v €(k) (W(e(k)))r " Ae!€a (pA,w(e(k)))

W (e(k)))rE

eAk)

AeA

(pA, W (£(k)))

> fw(e(k))

e eAk)

AeA

i+ e eAk) '

(13)

AeA

where r = (ri,... ,rn), rj = minrA. If E €

(k)

AeA

AeA

then the last term in (13) tends to infinity

also. So, we proved that mapping $ is proper. In spherical compactification of the space RN the closure of the orthant R^ is a manifold with a piecewise smooth boundary dR^. Consequently, the mapping $ transforms a manifold with boundary to a manifold with boundary.

As the mapping $ is proper, it follows that for any inner noncritical value x e (R+) full preimage $-1(x) being discrete and compact is a finite set. Hence one can define the degree of the mapping $ : i

^ RN as follows

deg*$ = E

sgn

d (x)

W).

(14)

The degree does not depend on the choice of noncritical value x G RN. To prove that we fix an arbitrary bounded neighborhood V C RN of the point x G RN and a set U such that V C $(U).

One can represent the degree degx$ by the following integral

$

degx$= / w($(£) - x), JdU

leu

where w is the Poincare form (see [8, II, Ch. 3]). This integral takes integer values and continuously depends on x G V therefore deg$ is a constant. So we proved that degree of the mapping $ does not depend on the choice of noncritical point x.

All that now remains to be shown is that deg$ = 1. In order to do that we use the following fact: the degree of the restriction $|dRN coincides with the degree deg$ (see [8, II, Ch. 3]). We

apply induction on dimension N. In the case N =1 the mapping $ is

X = e(1 + 0"r, 0 <r< 1

(15)

The function (15) is increasing so it implies that deg$ = 1. In the case N = 2 the restriction $|^A =o also has the form (15) therefore deg$|^A=o = 1. So the degree of the mapping $ : R+ ^

r

—»

R+ is equal to one also. For an arbitrary dimension N the restriction of the mapping $ on every face = 0} of the orthant R^ remains in the same class of the mappings. By inductive assumption we conclude that deg$ =1. □

We shall interpret the set A as a matrix A = (A«,..., A(n)) = (A\...,AW) , whose columns are the vectors Ak = (Ak,..., A^) of exponents of the monomials in (4). Here the ordering of the columns Ak inside each block A(i) is arbitrary but fixed.

We introduce two n x N matrices ' := DA, ' := ' — x, where x is the matrix whose ith row Xi is the characteristic function of the subset A(i) c A, that is, it has 1 on places A e A(i) and 0 elsewhere. We denote the rows of the matrices ' and ' by ..., and ^l,..., respectively. For any ordered set J = {j,..., jq} 1 < j < ... < jq < n, we fix the set of columns t = (rj1,..., Tjq) e AJ := A(jl) x ... x A(jq) and introduce q x q-matrix 'j(t). Denote by Aj (t ) the determinant of the matrix E — 'j (t ), where E is q-order unit matrix. The same determinants A/(t) arise in formula (10).

Theorem 2. The Mellin transform defined by the integral (11) is equal to " 1

M

where Q(z) is a polynomial of the form

n r((d<j),M) + (Vj,z)) r ((xj,z>)

(z) = H //-/ \---\-Q(z), (!6)

ji r( (d<j),M) + (^ ,z> + l)

Q(z) ((d(j),M) + (" ,z)j £ Aj (t )zT.

q=0 |j|=qj/J teAJ

The integral (11) converges under the following conditions

Re zA > 0, A e A, Re( ^ + ^ ^ z^) > 0, i = 1,...,n.

Proof. Theorem 1 argues validity of the change of variables (8) in the integral (11). Thus it follows from Lemma 1 that the integral (11) in coordinates £ looks like:

1 + Y T T Aj(t)£T | =

/»N I ^ ^ ^ j W»T I WD

/R+ \ q= i |j| =qteAJ J W

n

ni

c<j)z(i)-/dg(i) ^^ ^ . ( )

(s*),M>+<^i,z)+i + ^ aj(T)x

R+ W ' ' ' q=i |J| =q t£AJ

n „

n L

^J = i ii'-'i ! J (17)

(18)

-i-'«ia1" W(i,"'»>+(*'j 1(0,...,0), iej. 1 + Wi

All integrals in (17) may be calculated by the formula (see [9, formula 4.638(2)]):

f xz-/ dx r(zi) • ... • r(zq)r(s — zi — • • • — zq) JRI (1+ Xl +-----+ Xq)s r(s) .

In view of conditions Re s > 0, Re Zj > 0, i = 1,..., q on convergence of the integral (18) the integrals in (17) converge under the following conditions Re zA > 0, A G A, Re ^^d<j), ^ + (^j, z> j >

0, i = 1,..., n.

All that remains is to apply the formula (18) to the integrals in (17). So we obtain the required expression (16) for the Mellin transform. □

The first author's work was carried out with the financial support of RFBR (grant 11-0100852) and Ministry of Education and Science of Russian Federation (grant 1.34.11). The second author's work was carried out with the financial support of RFBR (grant 12-01-31021-mol_a).

References

[1] H.R.Mellin, Resolution de l'equation algebrique generale a l'aide de la fonction gamma, C.R. Acad. Sci, Paris Ser. I Math., 172(1921), 658-661.

[2] A.Yu.Semusheva, A.K.Tsikh, Continuation of Mellin's Studies on Solving Algebraic Equation , in Complex Analysis and Differential operators: On the Occasion of the 150th Anniversary of S.V. Kovalevskaya. Krasnoyarsk, 2000, 134-146 (in Russian).

[3] E.N.Mikhalkin, On solving general algebraic equations by integrals of elementary functions, Sib. Math. J., 47(2006), no. 2, 365-371.

[4] I.A.Antipova, A.K.Tsikh, The discriminant locus of a system of n Laurent polynomials in n variables, Izv. RAN. Ser. Math., 76(2012), no. 5, 29-56.

[5] I.A.Antipova, Inversion of many-dimensional Mellin transforms and solutions of algebraic equations, Matem. Sb., 198(2007), no. 4, 3-20.

[6] I.A.Antipova, An expression for the superposition of general algebraic functions in terms of hypergeometric series, Sib. Math. J., 44(2003), no. 5, 972-980.

[7] V.A.Stepanenko, On the solution of the system of n algebraic equations in n unknown variables in terms of hypergeometrical functions, Vestnik Krasnoyarskogo gosuniversiteta. Seriya phis.-mat. nauki, 2(2003), 35-48 (in Russian).

[8] B.A.Dubrovin, S.P.Novikov, A.T.Fomenko, Modern Geometry. Methods and Applications, Springer-Verlag, GTM 104, Part 2, 1985.

[9] I.S.Gradshteyn, I.M.Ryzhik, Table of Integrals, Series, and Products, Academic Press, Boston, MA, 1994.

Преобразование Меллина мономиальных функций решения общей полиномиальной системы

Ирина А. Антипова Татьяна В. Зыкова

В настоящей статье вычисляется преобразование Меллина мономиальной функции решения общей полиномиальной системы. При этом существенно используется линеаризация системы, которая в скалярном случае определяет биективную замену переменной. В случае системы уравнений требования к линеаризации ослаблены: она определяет собственное отображение, степень которого равна единице.

Ключевые слова: преобразование Меллина, алгебраическое уравнение.