Multivariate Newton Sums: Identities and Generating Functions

MULTIVARIATE NEWTON SUMS: IDENTITIES AND GENERATING FUNCTIONS Emmanuel Briand1 and Laureano Gonzalez-Vega2 1IRMAR, Universite ´ de Rennes I Rennes, France E-mail: [email protected] 2 Dpto. Matema ´ ticas, Estadstica y Computacio ´ n Universidad de Cantabria, Santander, Spain E-mail: [email protected] ABSTRACT This paper is devoted to present, first, a family of formulas extending to the multivariate case the classical Newton (or Newton–Girard) Identities relating the coefficients of a univariate polynomial equation with its roots through the Newton Sums and, secondly, the Generating Functions associated to the new introduced Newton Sums of the mul-tivariate case. As a by-product the kinds of systems accepting these Newton Identities are also characterized together with those allowing the Newton Sums to be computed in an inductive way directly from the coefficients of the polynomial system under consideration. INTRODUCTION When considering a univariate polynomial with coefficients in a field K with characteristic 0, its Newton Sums and its coefficients are connected by the well-known Newton Identities. Dealing with npolynomials in nunknowns with coefficients in a subfield Kof the field of the complex numbers C, and finitely many solutions in Cn, [1] showed that there exists a generalization of these formulas (Multivariate Newton Identities). They also pointed out that, when the system was a Pham System,i-e had the shape: Fi¼Xdi iþterms with total degree di1ð1inÞ;ð1Þ then these identities made possible the computation of the Multivariate Newton Sums from the coefficients, by an induction procedure, as in the univariate case. In a previous paper [8] it was shown how to obtain the same results with the algebraic version of the global residue operator (which made the method valid for systems with coefficients in any field Kwith characteristic 0). These results can be extended, in an algebraic way too, to the so-called Pham systems of type II (see also the analytic version in [1] ). In this paper we show that, using the same techniques introduced by [8] , the results extend to the case of Generalized Pham Systems,i-e systems: Fi¼Xdi iþlower terms w.r.t. ð1inÞð2Þ for some term ordering . This precisely means that, besides defining a zerodimensional complete intersection, the Fi’s are a Grobner basis for (see for instance [12] , Lemma 6.15). The vector d¼ðd1;...;dnÞis called the multidegree of the Generalized Pham System. In the first part we present the elements of the algebraic theory of the global residue operator to be used in the sequel. We also prove that when Fis a Generalized Pham System, the corresponding global residue operator maps exactly a polynomial on the leading coefficient of its normal form. This was already demonstrated in [5] , but our proof is direct. Besides, it leads naturally to a new proof of the Euler-Jacobi Theorem and Macaulay Theorem. Next we derive the Multivariate Newton Identities for Generalized Pham Systems F¼ðF1;...;FnÞand show how to use them as induction formulas to relate the generalized Newton sums (being defined most of them as the traces of monomials) with the coefficients of the polynomials Fi’s. The third part focuses on presenting the generating function for the generalized Newton sums introduced in the second section: i-e a rational function in the variables X1;...;Xnwhose expansion in negative powers of the variables provides the generalized Newton sums. This result is generalized to the case where, instead of the generalized Newton sums, we generate the image of the monomials under any linear form defined in the quotient K½X=hF1;...;Fni. This answers in a positive way a question posed by E. Becker. The last section shows how the previously mentioned Generating Functions for the Multivariate Newton Sums provide explicit formulas for the trace of some monomials with respect to a Pham System of degree 2. 1 ALGEBRAIC PRELIMINARIES Let Kbe a field of characteristic 0 and Lan algebraically closed field containing K. Let F1;...;Fnbe polynomials in K½X¼K½X1;...;Xn.We assume the ideal they generate, J, is a zero-dimensional complete intersection. Equivalently: the quotient algebra A¼K½X=Jis finite-dimensional over K. We denote by D1;...;Drthe roots of Jin Ln, and m1;...;mrtheir respective multiplicities. Then Ahas dimension D¼Pmi. This section focuses on the explicit presentation of some algebraic properties of Ato be used in the sequel. 1.1 The Dualizing Linear Form Since the ideal Jis a zero dimensional complete intersection, Ais a Frobenius Algebra [4] : there exists a linear form f:A!K(called a dualizing linear form) such that the symmetric K-bilinear form AA!K ða;bÞ 7! fðabÞ is non degenerate. [4] shows how to find such a dualizing linear form on A. We recall it now. For P2K½Xand 1 inwe define @iðPÞðX1;...;Xn;Y1;...;YnÞ¼ @iðPÞðX;YÞas follows: @iðPÞðX;YÞ¼PðY1;...;Yi1;Xi;...;XnÞPðY1;...;Yi;Xiþ1;...;XnÞ XiYi Then BezðX;YÞ, the Bezoutian Determinant associated to F1;...;Fn, is the determinant of the matrix of the @jðFiÞfor 1 i;jn(it is a polynomial in K½X;Y). Let A KAbe the quotient K½X;Y=hF1ðXÞ;...;FnðXÞ;F1ðYÞ;...; FnðYÞi and xiand yithe images of Xiand Yiin A KA. So Bezðx;yÞis the image in A KAof BezðX;YÞ. If B¼feiðxÞ:1iDgis a basis of the K-algebra Athen there exists a decomposition: Bezðx;yÞ¼b e1ðyÞe1ðxÞþþ b eDðyÞeDðxÞ: Then ^ B¼f b eiðxÞ:1iDghappens to be also a basis of A(the dual basis of B), and given the decomposition of 1 in Aaccording to the dual basis b B, 1¼d1b e1ðxÞþþdDb eDðxÞ, the linear form on Adefined by eiðxÞ 7! diis dualizing. This form ‘F¼‘½F1;...;Fnis called the Kronecker Symbol or global residue operator associated to F1;...;Fn. We will take the liberty of denoting also by ‘Fthe linear form it induces on K½X, when convenient. When KC, the Kronecker Symbol coincides with the classical residue (see [14] and [5] , (§ 0)). When Jis a radical ideal the global residue operator verifies: ‘FðPÞ¼X r i¼1 PðDiÞ JacðDiÞ where Jac denotes the Jacobian determinant of the polynomials F1;...;Fn. Without assumptions on Jthe dualizing form ‘Fis related with the trace by the following equality: ‘FðPJacÞ¼m1PðD1ÞþþmrPðDrÞ¼TraceðPÞ: A very useful tool to deal with the dualizing form is the Transformation Law which is presented in the following proposition ( [6] and [13] give an algebraic proof of this classical result). Proposition 1.1 ðTransformation LawÞ.Let G1;...;Gnbe polynomials in K½Xwith a finite number of solutions in Lnand Gi¼Ti1F1þþTinFn, Tij 2K½Xði2f1;...;ngÞ.IfP2K½Xthen ‘FðPÞ¼‘GðPdetðTÞÞ. 1.2 Expression of the Dualizing Form for a Generalized Pham System Now we consider the special case when Fis a Generalized Pham System for some term ordering . The family of the eaðxÞ¼xa1 1xan n¼ xa, for 0 a1<d1;...;0an<dn, is a basis over Kof the quotient algebra. We notice that we can write, for i6¼ j,@jðFiÞ¼bi;jðX;YÞ=Xjand @jðFjÞ¼ðXdj jþbj;jðX;YÞÞ=Xj where the bi;j’s, seen as polynomials in Xwith coefficients in K½Y, have all their monomials smaller than Xdi iwith respect to . Thus, if we develop the determinant defining the Bezoutian then we obtain: BezðX;YÞ¼ 1 X1Xn ðXd1 1Xdn nþRðX;YÞÞ with all X-monomials in Rsmaller than Xd1 1Xdn n. Otherwise stated: BezðX;YÞ¼Xd11 1Xdn1 nþR1ðX;YÞ with all X-monomials in R1smaller than Xd11 1Xdn1 n. The reduction of R1to its normal form with respect to F1ðXÞ;...; FnðXÞ;F1ðYÞ;...;FnðYÞnever produces the monomial Xd11 1Xdn1 n,so we get: Bezðx;yÞ¼xd11 1xdn1 nþX bi<di;i¼1...n xb^ ebðyÞ: Especially, ^ eðd11;...;dn1ÞðyÞ¼1, whence, for 0 a1<d1;...;0an<dn: ‘½F1;...;FnðxaÞ¼ 1ifa¼ðd11;...;dn1Þ 0 otherwise : We have just proved the following result which, for KC, is Lemma 4.2 of [5] . Proposition 1.2. Let ðF1;...;FnÞbe a Generalized Pham System for a term ordering , and Xdi ithe leading term of every Fi. Then ‘FðPÞis exactly the coefficient of Xd11 1Xdn1 nin the normal form of P 2K½Xwith respect to F1;...;Fnand . 1.3 Euler–Jacobi Theorem and Macaulay’s Theorem For H2K½X, its Newton Diagram NDðHÞZnis the set of the exponents of the monomials in Hwith non-zero coefficient. The Newton Polytope of H, denoted with NPðHÞ, is the convex hull of NDðHÞin Rn. The Newton Polytope of the product F1Fnis also the Minkowski sum of the NPðFiÞ’s: NPðF1FnÞ ¼ NPðF1ÞþþNPðFnÞ: For EZn, we denote the semi-group generated by E(the set of all the finite sums e1þþemwith the ei2E)withPE. We begin by looking at the special case when ðF1;...;FnÞis a Generalized Pham System, with Xd1 1;...;Xdn nthe respective leading terms of F1;...;Fn,and d¼ðd1;...;dnÞ.LetSðFÞbe the semi-group generated by the elements of the set fdþaja2[ n i¼1NDðFiÞg.ThissetSðFÞis always contained in CðFÞ¼fdtx:t2Rþ;x2NPðF1FnÞg the polyhedron obtained by drawing half-lines from dthrough all points of NðF1FnÞ, next translating it by dand finally rotating it by half a turn around 0. The origin 0 is a vertex for C, and for any linear form win the normal cone of NPðF1FnÞat d, the hyperplane w¼0 is a supporting hyperplane of CðFÞat 0. When all the Fi’s coefficients are non-zero and have independent parameters then SðFÞ¼CðFÞ\Zn. Proposition 1.3. Let F be a Generalized Pham System. If P 2K½Xand NDðPÞlies outside fd1gþSðFÞthen ‘FðPÞ¼0. Here 1 is the vector ð1;...;1Þ2Zn. Proof. Suppose Preduces through F1, for instance, into a polynomial Q. Then: NDðQÞ  NDðPÞ þ NDðF1Þ þ fðd1;0;...;0Þg More generally, if Ris obtained from Pafter a series of reductions by F1;...;Fn, we’ll have: NDðRÞNDðPÞþXNDðF1Þ þ fðd1;0;...;0Þgþ þXNDðFnÞþfð0;...;0;dnÞg: Note that actually: XNDðF1Þ þ fðd1;0;...;0Þgþ þXNDðFnÞþfð0;...;0;dnÞg¼SðFÞ for every set PðNDðFiÞþfð...;0;di;0;...ÞgÞ containing f0g. After sufficiently many reductions, Rhappens to be the normal form of P. Its global residue is non-zero if and only if its Newton Diagram contains fd1g. This would involve that fd1g 2 NDðPÞ þ ðSðFÞÞ whence the proposition. j Example 1.1. A very special case is when the polynomials are pure powers: F1¼Xd1 1;...;Fn¼Xdn n. Then the global residue operator acts on K½Xas follows: ‘½Xd1 1;...;Xdn nðXaÞ¼ 1ifa¼ðd11;...;dn1Þ 0 otherwise : Let us introduce some vocabulary. Let w2Nn(a weight vector). It determines a linear form on Rn: the weight w:ðt1;...;tnÞ 7! w1t1þ þwntn. The w-degree of a monomial Xais defined as wðaÞ. The w-degree of a non-zero polynomial Pis the integer m¼degwðPÞsuch that P¼HþR with all monomials in Hhaving w-degree m, and all monomials in Rhaving w-degrees lower than m. The polynomial His then the w-homogeneous leading part of P. A root of H, if different from the trivial root ð0;...;0Þ,is said to be a root at the weighted infinity, with respect to w, for P. If F1;...;Fnhave no common root at the weighted infinity with respect to w, they are said to define a strict complete intersection graded by w. Strict complete intersections are studied in [10] and in [5] . Example 1.2. A General Pham System Fdefines a strict complete intersection, graded by any weight wlying inside the normal cone of NPðF1 FnÞat d, where dis the multidegree of F. Conversely, if the system F¼ðF1;...;FnÞdefines a strict complete intersection graded by wand such that the leading w-homogeneous parts of F1;...;Fnare pure powers, Xd1 1;...;Xdn n, then it is a Generalized Pham System for any term ordering compatible with win the following sense: wðXaÞ<wðXbÞ)XaXb. One can construct such an ordering by choosing an arbitrary term ordering Tand setting: XaXb() wðXaÞ<wðXbÞorwðXaÞ¼wðXbÞand XaTXb: With a classical method and using the Transformation Law, the study of ‘Fin the zero-dimensional strict complete intersection case is reduced to the case of Generalized Pham Systems. Suppose F1;...;Fndefine a zerodimensional strict complete intersection graded by w. Let H1;...;Hnbe the respective w-homogeneous leading parts of F1;...;Fn. Then, for r1;...;rn2 Nlarge enough, there exist polynomials Tij such that: Xwiri i¼Ti1ðXwÞH1ðXwÞþþTinðXwÞHnðXwÞð1inÞð3Þ where Xwstands for Xw1 1;...;Xwn n. We can assume that the Tij’s are w-homogeneous (of respective w-degrees wiridegwðFjÞ). Then (i2f1;...;ng): Gi¼Ti1F1þþTin Fnð4Þ is a Generalized Pham System and ‘FðPÞ¼‘GðPdetðTÞÞ . Remark 1.1. Note that detðTÞis w-homogeneous of w-degree PwiriPdegwðFiÞ. Theorem 1.1 ðWeighted version of the EulerJacobi TheoremÞ.Let w be a weight vector, kwk¼w1þþwn, and F1;...;Fnwith no common solution at the weighted infinity with respect to w.IfP2K½Xand degwðPÞ<X n i¼1 degwðFiÞkwk then we have ‘FðPÞ¼0. The integer PidegwðFiÞkwkis called the critical number associated to Ffor w. Proof. First, the result is obvious when Fis a Generalized Pham System, with multidegree d, because w¼PdegwðFiÞkwkis a supporting hyperplane of fd1gþCðFÞat d1. Now, if Fjust defines a zero-dimensional complete intersection, we set Tand Gas in (3), (4). If degwðPÞ<PdegwðFiÞkwk, then degwðP detðTÞÞ <Pwirikwk(by Remark 1.1). This last number is exactly the critical number of the system Gfor w. The theorem follows. j Theorem 1.2 ðWeighted version of Macaulay0s TheoremÞ.Let w be a weight vector, and H1;...;Hnbe w-homogeneous polynomials, with no common nontrivial zero. Let P be a w-homogeneous polynomial. If degwðPÞ> PdegwðHiÞkwkthen P lies in the ideal hH1;...;Hni. Proof. We choose r1;...;rnand Tas in (3). Then, by Example 1.1, ‘HðPÞ¼‘½Xr1 1;...;Xrn nðPdetðTÞÞ ¼ 0 as soon as NDðPÞdoes not meet ðr11;...;rn1Þ  NDðdetðTÞÞ. This last set is contained in the hyper- plane w¼PdegwðHiÞkwk.SoifPis homogeneous of degree larger than this number, then ‘HðPÞ¼0, and better: ‘HðPQÞ¼0 for every Q2K½X. But this implies Plying in the ideal hH1;...;Hni, because the global residue operator is dualizing. j Remark 1.2. As noticed by [3] and [5] , Macaulay’s theorem provides convenient r1;...;rnfor the construction (3), (4): it is sufficient to solve a linear system. Example 1.3. Let w¼ð1;2Þand F1¼X2 1X2 2þX2 1þX2 2þX1X2þX1þ1; F2¼X4 1þX2 2X2 1X2þX1þ1: Then F1;F2have weighted degrees, with respect to w, repectively 6;4, and no common solution at the weighted infinity. The system has critical number 7. So X8 1and X4 2are in the ideal generated by the w-homogeneous leading parts X2 1X2 2and X4 1þX2 2X2 1X2. Looking for a1;a2;b1;b2;b3;u1;u2;v1;v2;v3such that: X8 1¼ða1X2 1þa2X2ÞX2 1X2 2þðu1X4 1þu2X2 1X2þb3X2 2ÞðX4 1þX2 2X2 1X2Þ X4 2¼ðu1X2 1þu2X2ÞX2 1X2 2þðv1X4 1þv2X2 1X2þv3X2 2ÞðX4 1þX2 2X2 1X2Þ we find easily there is a unique solution: a1¼v1¼v2¼b3¼0;a2¼u1¼ 1;b1¼b2¼u2¼v3¼1 and: T¼X2X4 1þX2 1X2 X2 1þX2X2 2  : The Generalized Pham System G¼TFis: G1¼X8 1X23X1X22X1X2X2þX15þX14þX13X2 G2¼X4 2X14X12X22X13X2X13X12þX2X12 þX23þ2X1X22þX1X2þX2þX22: In Fig. 1, the shaded polytope is fd1g þ NPðG1G2Þ, and the shaded polyhedron is fd1gþCðGÞ. The weighted Euler–Jacobi theorem says the ‘GðXaÞ¼0 for abelow the dotted line w¼13, but Proposition 1.3 implies this for the cases in the broader white area. F1FnX g2Zn ~ ‘FðXgPÞ Xg¼PX1Xn:ð9Þ At this point we guess this may lead to generating functions for the family of the ~ ‘FðXgPÞ;g2Zn, and especially for the Generalized Newton Sums, with P¼Jac. The following lemma will give the appropriate setting for these considerations. Let us introduce the necessary notions: let wbe a weight and Bwbe the set of all expressions Pg2Zn ug Xgwith ug2K, and such that 1. for any k2Z, all but a finite number of the ugwith wðgÞ¼kare zero, and 2. there exists k02Zsuch that ug¼0 as soon as wðgÞ<k0. This is naturally a graded K-algebra. If Pis a polynomial whose whomogeneous leading part is a monomial, so P¼XdþRwith wðRÞ<wðdÞ, then the formal series S¼X i2NR Xdi is well-defined in Bw,andPS¼1. In this way in our ring Bwall the inverses of such polynomials Pare represented. We say that Sis the series expansion of 1=Pin Bw. Remark 3.1. If KC, we define: Log :Cn!Rn ðz1;...;znÞ 7!ðLogjz1j;...;LogjznjÞ and Ndas the normal cone of Pat d. Then the series Sconverges in Log1ðbþNdÞfor some b2Rn(see [7] ). Lemma 3.1. Let F1;...;Fnbe a Generalized Pham System, with multidegree d. With the notations of Lemma 2:1: the series X g2Zn ~ ‘FðXgPÞ Xg lies in Bw, for any weight wlying inside the normal cone of NPðF1FnÞat d, and: X g2Zn ~ ‘FðXgPÞ Xg¼PX1Xn F1Fn in Bw. Proof. The series is in Bwbecause of Proposition 2.1, 2. Its expression as a rational fraction comes from the above calculation. j Suppose we have a zero-dimensional system F¼ðF1;...;FnÞ, which can be transformed into a Generalized Pham System G(1 in): Gi¼Ti1F1þþTin Fn:ð10Þ For g2Nn:‘FðXgPÞ¼‘GðXgPdetðTÞÞ. So: X g2Nn ‘FðXgPÞ Xg¼PdetðTÞX1Xn G1Gn þX g62Nn mg Xgð11Þ for some mg. Precisely, mg¼~ ‘GðXgPdetðTÞÞ. Example 3.1. If Fdefines a zero-dimensional strict complete intersection, it can be transformed as above into a Generalized Pham System, with the method used in the Proof of 1.3, and Macaulay’s Theorem (see Remark 1.2). This was pointed out in [2] and [5] . In these papers the equality (15) was used to compute the residues by series expansions. Example 3.2. For Fdefining a zero-dimensional complete intersection, there always exist ‘‘eliminating equations’’: Ti1F1þþTin Fn¼wiðXiÞð12Þ for i¼1...n,withTij 2K½Xand the wi2K½Xinon-zero. We get: X a2Nn Sa Xa¼detðTÞJac X1Xn w1wn þX a62Nn ma Xa for some ma,a62 Nn. This result, for n¼2, had been established by Jacobi in 1835 in the particular case when every wiis the Eliminant polynomial of Ffor Xi(that is: wihas no multiple root, and its roots are precisely the Xicoordinates of the roots of F) [9] . Much later, in 1991, P. Pedersen proved the general case [11] . Now we find what are precisely the generating series for the Newton Sums (without terms mawith a62 Nn), and for the families defined in a similar way, that is: the families of all of the lðXaÞfor a given linear form l on A. Note that, since ‘Fis dualizing, there exists a polynomial Pl(unique modulo hF1;...;Fni) such that l:Q7! ‘FðQPlÞ. Proposition 3.1. Suppose F1;...;Fndefine a zero-dimensional complete intersection. Let f by any K-linear form on A. Then the generating function of the fðXaÞis obtained by series expansion in 1=X1;...;1=Xnof a rational function: X a2Nn fðXaÞ Xa¼X1Xn w1ðX1ÞwnðXnÞNFwðdetðTÞPlÞ where T is a transformation matrix of F into wðas in 16Þand NFwgives the normal form with respect to w1;...;wn. Proof. For a2Nn:‘wðXaNFwðdetðTÞPfÞÞ ¼ ‘wðXadetðTÞPfÞ¼fðXaÞ. And for a62 Nn: ‘wðXaNFwðdetðTÞPfÞÞ ¼ 0 because the wi’s are univariate (see Example 2.2). j Example 3.3. This formula produces some well-known identities for a single univariate polynomial PðXÞ: With f¼Trace, the fðXiÞare the Newton Sums siassociated to P, and X i2N si Xi¼XP0 P: With f¼‘P, the global residue operator: X i2N ‘PðxiÞ Xi¼X P: 4 AN APPLICATION: PHAM SYSTEMS OF DEGREE 2 In the particular case when all the Fihave degree 2 and the system is a Pham system (then we denote: Fi¼X2 iþPn j¼1ui;jXjþvi;i¼1;...;n), the Newton Sums S1;1;...;1;0;0;...;0have quite a simple expression as functions of the coefficients. Indeed, for such a system, developing in Laurent series the function X1XnJac F1Fn gives: Sa¼X M;I;J ð1ÞjMj2CardIY n i¼1 mi;0þþmi;n mi;0;...;mi;n  detUðJÞY n i¼1 vmi;0 iY n i;j¼1 umi;j i;j where the sum is carried over all of the decompositions I]J¼f1;...;ng and matrices M¼ðmi;jÞ;i¼1;...;n;j¼0;...;nwith positive entries, such that: ai¼X j¼0;...;n ml;jX i¼1;...;n mi;lwIðiÞ;i¼1;...;n where wIðiÞ¼1ifi2I, 0 otherwise. Our notation UðJÞstands for the submatrix of the ui;jwith i2J;j2J. For a¼ð1;...;1;0;...;0Þ:ktimes 1 followed by nktimes 0, these I;J;Mshould verify: 2X j¼0;...;n ml;jX i¼1;...;n mi;l¼ 1ifikand i2I 1ifi>kand i2J 0 otherwise (: Adding these equalities together one gets: jMjþX n i¼1 mi;0¼Cardðf1;...;kg\IÞCardðfkþ1;...;ng\JÞ: But obviously everytime ikand i2Ithen the i-th line of Mhas a nonzero entry, so jMjCardðf1;...;kg\IÞ. From this we deduce that for all i, mi;0¼0; Jf1;...;kg; and finally that the matrix of the mi;j;i2I;j2Iis a permutation matrix (all coefficients equal to 0 except precisely one 1 on each column and one 1 on each row) and the remaining coefficients mi;j¼0 for i62 Ior j62 I. This Newton Sum has an expression of the following type: S1;...;1;0...;0¼2nkX J]L¼f1;...;kg;t2SL ð2ÞCardðLÞdetðUðJÞÞ  Y i2L ui;tðiÞ ¼ð1Þk2nkX s2Sk tðsÞY k i¼1 ui;sðiÞ where tðsÞ¼ð2l11Þð2lp1Þwhen sdecomposes as a product of p cycles with disjoint supports of respective lengths l1;...;lp. ACKNOWLEDGMENTS We thank E. Becker for his comments and support for this work since he proposed to us the question about the existence of generating functions for the Multivariate Newton Sums and other kinds of similar multivariate symmetric functions. Partially supported by the grant DGESIC PB 98-0713- C02-02 (Ministerio de Educacio ´n y Cultura). REFERENCES 1. Aizenberg, L.A.; Kytmanov, A.M. Multidimensional Analogues of Newton’s Formulas for Systems of Nonlinear Algebraic Equations and some of their Applications. Sibirskii Matematicheskii Zhurnal 1981,22 (2), 19–39. English translation in Siberian Mathematical J. 1981,22, 180–189. 2. Aizenberg, L.A.; Yuzhakov, A. 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