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.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

Antipova, Irina A.

Zykova, Tatyana V.

Антипова, Ирина А.

Зыкова, Татьяна В.

Siberian Federal University Digital Repository

Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156УДК 517.55Mellin Transform for Monomial Functions of the Solution tothe General Polynomial SystemIrina A. Antipova∗Institute of Space and Information Technology,Siberian Federal University,Kirensky, 26, Krasnoyarsk, 660074RussiaTatyana V. Zykova†Institute of Space and Information Technology,Siberian Federal University,Kirensky, 26, Krasnoyarsk, 660074RussiaReceived 10.12.2012, received in revised form 10.01.2013, accepted 20.01.2013In 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 caseit deﬁnes bijective change of variables. In case of the system of equations we weaken requirements on themapping given by the linearization: it is proper and its degree is equal to one.Keywords: Mellin transform, algebraic equation.IntroductionConsider a polynomial mapP = (P1, . . . , Pn) : Cn → Cn.We assume that the sets A(i) ⊂ Zn+ of exponents of the monomials in polynomials Pi are ﬁxedwhile 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∑λ∈A(i)a(i)λ yλ = 0, i = 1, . . . , n (1)with unknown y = (y1, . . . , yn) ∈ Cn and variable coeﬃcients a(i)λ , where A(i) ⊂ Zn+ are ﬁxedﬁnite subsets, λ = (λ1, . . . , λn), yλ = yλ11 . . . yλn1 .In 1921 Mellin [1] presented an integral formula and series expansion for the positive poweryµ(x) of the function y(x) deﬁned by the general (reduced) algebraic equationym + x1yλ1 + · · ·+ xpyλp − 1 = 0, (2)∗antipova@akadem.ru†zykovatv@mail.ruc© Siberian Federal University. All rights reserved– 150 –Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156m > λ1 > · · · > λp > 1. This result was extensively applied by the Krasnoyarsk School (see, forexample [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 function1yµ(−x):=1yµ11 (−x) · · · yµnn (−x), µ = (µ1, . . . , µn) , µi > 0, (3)composed of coordinates of the solution y(x) to the reduced polynomial system of the formyd(i)+∑λ∈Λ(i)x(i)λ yλ − 1 = 0, i = 1, . . . , n, (4)where the matrix of the columns of distinguished exponents(d(1), . . . , d(n))=: D is non-degenerate and Λ(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 diagonalelements m1, . . . ,mn. Let Λ be the disjunctive union of Λ(i), and let N be the cardinality ofΛ, i.e., the number of coeﬃcients in system (4). The set of these coeﬃcients is a vector spaceCΛ ∼= CΛx , where the coordinates of points x = (xλ) are indexed by the elements λ ∈ Λ. We usu-ally distinguish the group of coordinates corresponding to the indices λ ∈ Λ(i) by writing x(i).Let ♯Λ(i) be the cardinality of the set Λ(i). We will assume that all sets Λ(i) lie in the interior ofthe simplex with vertices 0, m1e1, . . . ,mnen, where e1, . . . , en is a basis in Zn. In other words allpoints λ ∈ Λ have nonzero coordinates and satisfy the following conditionn∑i=1〈d˜(i), λ〉< 1, (5)where d˜(i) are columns of the matrix D˜ := D−1.In Section 1 we discuss the deﬁnition and the Jacobian of linearization (ξ,W ) → (x, y)of system (4) which is the main tool in calculations of the Mellin transform for the monomialfunction (3). Section 2 contains the main results of the present paper. The ﬁrst one is Theorem 1which states that the mapping Φ deﬁned by the linearization (ξ,W ) → (x, y) is proper and itsdegree degΦ is equal to one. The second one is Theorem 2 which gives the Mellin transform forthe 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 spaceCΛ × Tn with coordinates x =(x(i)λ), y = (y1, . . . , yn) and introduce a change of variables(ξ,W )→ (x, y) in CΛ × Tn by puttingx(i)λ = −ξ(i)λ W−D˜λ, λ ∈ Λ(i), i = 1, . . . , n,y = W D˜,(6)where W = (W1, . . . ,Wn) , W−D˜λ =n∏i=1W−〈d˜(i),λ〉i , WD˜ =(W d˜(1), . . . ,W d˜(n)), ξ =(ξ(i)λ).Therefore we can write (4) as a system of linear equationsWi = 1 +∑λ∈Λ(i)ξ(i)λ , i = 1, . . . , n. (7)– 151 –Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156If we deﬁne a change of variables ξ → x in the space of coeﬃcients CΛ by the formulax(i)λ = ξ(i)λn∏k=11 + ∑τ∈Λ(k)ξ(k)τ−〈d˜(k),λ〉, λ ∈ Λ(i), i = 1, . . . , n, (8)then vector y (−x(ξ)) takes the following formy (−x(ξ)) = (W (ξ))D˜, (9)where W (ξ) =(W1(ξ(1)), . . . ,Wn(ξ(n)))is formed from the linear functions (7).The idea of linearizing an algebraic equation by using a change of variables of this type isdue to Mellin. It was realized by Mellin in [1] (see also [5]) in order to obtain the integralrepresentation and series expansion for the solution of algebraic equation (2). In [6] this trickwas applied to the "triangular" system when the ﬁrst equation depends only on y1, the secondone 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 typeyµ(x) := yµ11 (x) · · · yµnn (x), µ = (µ1, . . . , µn) , µi > 0.The main result of [7] is the power series expansion of the function yµ(x). This expansion wasgiven using the formal calculation of the Mellin transform for M [yµ(x)] .Further we need to consider the restriction Φ of the mapping CNξ → CNx given by (8) on thepositive orthant RN+ .Lemma 1. The Jacobian of the mapping Φ is equal to∂(x)∂(ξ)=n∏i=1W−∑λ∈Λ(i)〈d˜(i),λ〉−1i1 +n∑q=1∑|I|=q∑τ∈ΛI∣∣∣∣∣∣∣∣∣1−τi1i1mi1. . . −τi1iqmi1. . . . . . . . .−τiqi1miq. . . 1−τiqiqmiq∣∣∣∣∣∣∣∣∣ξτ , (10)where I is an odered set 1 6 i1 < · · · < iq 6 n, τ = (τi1 , . . . , τ iq ) is an element of Cartesianproduct ΛI = Λ(i1) × · · · × Λ(iq), and ξτ =∏i∈Iξτ i .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 diagonalblock are of the following type∂x(i)λ∂ξ(i)λ=(1−〈d˜(i), λ〉ξ(i)λ W−1i)W−D˜λ;the elements outside the diagonal in this block are∂x(i)λ∂ξ(i)τ= −〈d˜(i), λ〉ξ(i)λ W−1i W−D˜λ.The nondiagonal (i, k)-block contains the following elements∂x(i)λ∂ξ(k)τ= −〈d˜(i), λ〉ξ(i)λ W−1i W−D˜λ, k 6= i.– 152 –Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156The computation gives us∂(x)∂(ξ)=n∏i=1W−∑λ∈Λ(i)〈d˜(i),λ〉−1i detδjkWj − ∑λ∈Λ(k)〈d˜(j), λ〉ξ(k)λj,k=1,n,where δjk is the Kronecker symbol, j, k = 1, . . . , n.The last determinant may be reduced by the formula:det(E +A) = 1 +n∑q=1∑|I|=q∣∣∣∣∣∣∣Ai1i1 . . . Ai1iq. . . . . . . . .Aiqi1. . . Aiqiq∣∣∣∣∣∣∣ ,where E is an unit matrix, A = (Aji ) is an arbitrary matrix of order n, and I is an ordered set1 6 i1 < . . . < iq 6 n.All that now remains to be written is that∂(x)∂(ξ)=n∏i=1W−∑λ∈Λ(i)〈d˜(i),λ〉−1i1 + n∑q=1∑|I|=q∣∣∣∣∣∣∣Ai1i1 . . . Ai1iq. . . . . . . . .Aiqi1. . . Aiqiq∣∣∣∣∣∣∣ ,where Airil =∑λ∈Λ(ir)(δrl −〈d˜(il), λ〉)ξ(ir)λ , r, l = 1, . . . , q, and δrl is the Kronecker symbol. Hencewe get the desired formula (10). 2. Mellin transform for the function 1yµ(−x)We consider the monomial function (3) where yj(−x) are branches with conditions yi(0) = 1,i = 1, . . . , n. Let us recall that Mellin transform of the function 1yµ(−x) is deﬁned by the followingintegralM[1yµ(−x)](z) =∫RN+1yµ(−x)xz−Idx, (11)where xz−I = xz1−11 · · ·xzN−1N , dx = dx1 · · · dxN (see, for example [5]). To calculate the integral(11) we consider the transformation ξ → x (or a mapping Φ : RN+ → RN+ ) given by (8).Theorem 1. The mapping Φ is proper. Its degree degΦ is correctly defined and degΦ = 1.Proof. We prove that Φ(∂RN+ ) = ∂RN+ . Further we miss the upper index in the notationsx(i)λ , ξ(i)λ , and write xλ, ξλ for simplicity. Note that any coordinate plane ξλ = 0 is mapped tocoordinate plane xλ = 0. Moreover, the boundary points of the orthant are mapped only to theboundary points. If a sequence ξ(k) ∈ RN+ , k ∈ N, converges to the boundary point ξ ∈ ∂RN+then the sequence of images Φ(ξ(k)), k ∈ N, also converges to a boundary point of the orthant.We are now going to prove that condition ξ(k) → +∞ implies x(k) = Φ(ξ(k))→ +∞. Note thatcoordinate x(k)λ may be ﬁnite in case when the corresponding coordinate ξ(k)λ tends to +∞, but∣∣∣x(k)∣∣∣ := ∑λ∈Λx(k)λ → +∞– 153 –Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156when ∣∣∣ξ(k)∣∣∣ = ∑λ∈Λξ(k)λ → +∞.Using assumption (5) for any λ ∈ Λ one can choose such a real positive n-dimensional vectorsrλ = (rλj ), pλ = (pλj ) that 〈d˜(j), λ〉+ rλj = pλj , |pλ| = 1. (12)Using the well-known Jensen inequalityap11 . . . apnn 6 p1a1 + · · ·+ pnan,which is valid for any positive numbers a1, . . . , an, p1, . . . , pn,∑pi = 1, and conditions (12) weget the following estimates:∑λ∈Λx(k)λ =∑λ∈Λξ(k)λ(W (ξ(k)))−D˜λ=∑λ∈Λξ(k)λ(W (ξ(k)))rλ (W (ξ(k)))−pλ>>∑λ∈Λξ(k)λ(W (ξ(k)))rλ〈pλ,W (ξ(k))〉 > (W (ξ(k)))r∑λ∈Λξ(k)λ〈pλ,W (ξ(k))〉 > (W (ξ(k)))r∑λ∈Λξ(k)λ1 +∑λ∈Λξ(k)λ, (13)where r = (r1, . . . , rn) , rj = minλ∈Λrλj . If∑λ∈Λξ(k)λ → +∞, then the last term in (13) tends to inﬁnityalso. So, we proved that mapping Φ is proper. In spherical compactiﬁcation of the space RN theclosure of the orthant RN+ is a manifold with a piecewise smooth boundary ∂RN+ . Consequently,the mapping Φ transforms a manifold with boundary to a manifold with boundary.As the mapping Φ is proper, it follows that for any inner noncritical value x ∈(RN+)ofullpreimage Φ−1(x) being discrete and compact is a ﬁnite set. Hence one can deﬁne the degree ofthe mapping Φ : RN+ → RN+ as followsdegxΦ =∑ξ:Φ(ξ)=xsgn∂(x)∂(ξ). (14)The degree does not depend on the choice of noncritical value x ∈ RN+ . To prove that we ﬁx anarbitrary bounded neighborhood V ⊂ RN+ of the point x ∈ RN+ and a set U such that V ⊂ Φ(U).One can represent the degree degxΦ by the following integraldegxΦ =∫∂Uω(Φ(ξ)− x),where ω is the Poincare´ form (see [8, II, Ch. 3]). This integral takes integer values and continu-ously depends on x ∈ V therefore degΦ is a constant. So we proved that degree of the mappingΦ does not depend on the choice of noncritical point x.All that now remains to be shown is that degΦ = 1. In order to do that we use the followingfact: the degree of the restriction Φ|∂RN+ coincides with the degree degΦ (see [8, II, Ch. 3]). Weapply induction on dimension N . In the case N = 1 the mapping Φ isx = ξ(1 + ξ)−r, 0 < r < 1. (15)The function (15) is increasing so it implies that degΦ = 1. In the case N = 2 the restrictionΦ|ξλ=0 also has the form (15) therefore degΦ|ξλ=0 = 1. So the degree of the mapping Φ : R2+ →– 154 –Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 150–156R2+ is equal to one also. For an arbitrary dimension N the restriction of the mapping Φ onevery face {ξλ = 0} of the orthant RN+ remains in the same class of the mappings. By inductiveassumption we conclude that degΦ = 1. We shall interpret the set Λ as a matrix Λ =(Λ(1), . . . ,Λ(n))=(λ1, . . . , λN), whose columnsare the vectors λk =(λk1 , . . . , λkn)of exponents of the monomials in (4). Here the ordering of thecolumns λk inside each block Λ(i) is arbitrary but ﬁxed.We introduce two n × N matrices Ψ := D˜Λ, Ψ˜ := Ψ − χ, where χ is the matrix whose ithrow χi is the characteristic function of the subset Λ(i) ⊂ Λ, that is, it has 1 on places λ ∈ Λ(i)and 0 elsewhere. We denote the rows of the matrices Ψ and Ψ˜ by ψ1, . . . , ψn and ψ˜1, . . . , ψ˜nrespectively. For any ordered set J = {j1, . . . , jq} 1 6 j1 < . . . < jq 6 n, we ﬁx the set ofcolumns τ =(τ j1 , . . . , τ jq)∈ ΛJ := Λ(j1)× . . .×Λ(jq) and introduce q× q-matrix ΨJ(τ). Denoteby ∆J (τ) the determinant of the matrix E −ΨJ (τ), where E is q-order unit matrix. The samedeterminants ∆I(τ) arise in formula (10).Theorem 2. The Mellin transform defined by the integral (11) is equal toM[1yµ(−x)](z) =n∏j=1Γ(〈d˜(j), µ〉+〈ψ˜j , z〉)Γ (〈χj , z〉)Γ(〈d˜(j), µ〉+ 〈ψj , z〉+ 1) Q(z), (16)where Q(z) is a polynomial of the formQ(z) =n∑q=0∑|J|=q∏j /∈J(〈d˜(j), µ〉+〈ψ˜j , z〉) ∑τ∈ΛJ∆J(τ)zτ .The integral (11) converges under the following conditionsRe zλ > 0, λ ∈ Λ, Re(〈d˜(i), µ〉+〈ψ˜i, z〉)> 0, i = 1, . . . , n.Proof. Theorem 1 argues validity of the change of variables (8) in the integral (11). Thus itfollows from Lemma 1 that the integral (11) in coordinates ξ looks like:∫RN+1 + n∑q=1∑|J|=q∑τ∈ΛJ∆J (τ)ξτ ξz−I dξW D˜µ+Ψz+I==n∏i=1∫R♯Λ(i)+ξ(i)z(i)−Idξ(i)W〈d˜(i),µ〉+〈ψi,z〉+1i+n∑q=1∑|J|=q∑τ∈ΛJ∆J (τ)××n∏i=1∫R♯Λ(i)+ξ(i)z(i)−I+δiJ dξ(i)W〈d˜(i),µ〉+〈ψi,z〉+1i, δiJ ={(1, . . . , 1), i ∈ J,(0, . . . , 0), i /∈ J.(17)All integrals in (17) may be calculated by the formula (see [9, formula 4.638(2)]):∫Rq+xz−I dx(1 + x1 + · · ·+ xq)s=Γ(z1) · ... · Γ(zq)Γ(s− z1 − · · · − zq)Γ(s). (18)In view of conditions Re s > 0, Re zi > 0, i = 1, . . . , q on convergence of the integral (18) the inte-grals in (17) converge under the following conditions Re zλ > 0, λ ∈ Λ, Re(〈d˜(i), µ〉+ 〈ψi, z〉)>0, i = 1, ..., n.– 155 –Irina A.Antipova, Tatyana V.Zykova, Mellin transform for monomial functions of the solution...All that remains is to apply the formula (18) to the integrals in (17). So we obtain therequired expression (16) for the Mellin transform. The first author’s work was carried out with the financial support of RFBR (grant 11-01-00852) and Ministry of Education and Science of Russian Federation (grant 1.34.11). The secondauthor’s work was carried out with the financial support of RFBR (grant 12-01-31021-mol a).References[1] H.R.Mellin, Re´solution de l’e´quation alge´brique ge´ne´rale a` l’aide de la fonction gamma, C.R.Acad. Sci., Paris Se´r. I Math., 172(1921), 658-661.[2] A.Yu.Semusheva, A.K.Tsikh, Continuation of Mellin’s Studies on Solving Algebraic Equa-tion , in Complex Analysis and Differential operators: On the Occasion of the 150th An-niversary 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 inn variables, Izv. RAN. Ser. Math., 76(2012), no. 5, 29–56.[5] I.A.Antipova, Inversion of many-dimensional Mellin transforms and solutions of algebraicequations, Matem. Sb., 198(2007), no. 4, 3-20.[6] I.A.Antipova, An expression for the superposition of general algebraic functions in termsof 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 unknownvariables 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 реше-ния общей полиномиальной системыИрина А. АнтиповаТатьяна В. ЗыковаВ настоящей статье вычисляется преобразование Меллина мономиальной функции решения об-щей полиномиальной системы. При этом существенно используется линеаризация системы, ко-торая в скалярном случае определяет биективную замену переменной. В случае системы урав-нений требования к линеаризации ослаблены: она определяет собственное отображение, степенькоторого равна единице.Ключевые слова: преобразование Меллина, алгебраическое уравнение.– 156 –

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

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