To solving multiparameter problems of algebra. 9. The Ψ<Emphasis Type="Italic">F-q</Emphasis> method for factorizing invariant polynomials and its applications

A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173.     

To Solving Multiparameter Problems of Algebra. 4. The AB-Algorithm and its Applications

The paper continues the investigation of methods for factorizing q-parameter polynomial matrices and considers their applications to solving multiparameter problems of algebra. An extension of the AB-algorithm, suggested earlier as a method for solving spectral problems for matrix pencils of the form A - λB, to the case of q-parameter (q ≥ 1) polynomial matrices of full rank is proposed. In accordance with the AB-algorithm, a finite sequence of q-parameter polynomial matrices such that every subsequent matrix provides a basis of the null-space of polynomial solutions of its transposed predecessor is constructed. A certain rule for selecting specific basis matrices is described. Applications of the AB-algorithm to computing complete polynomials of a q-parameter polynomial matrix and exhausting them from the regular spectrum of the matrix, to constructing irreducible factorizations of rational matrices satisfying certain assumptions, and to computing “free” bases of the null-spaces of polynomial solutions of an arbitrary q-parameter polynomial matrix are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 127–143.     

An approach to solving multiparameter algebraic problems

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on q≥2 spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the GCD of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called ΔW-q factorizations of polynomial q-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1–3] to an arbitrary q≥2. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 191–246. Translated by V. N. Kublanovskaya.     

To solving multiparameter problems of algebra. 7. The PG-q factorization method and its applications

The paper continues the development of rank-factorization methods for solving certain algebraic problems for multi-parameter polynomial matrices and introduces a new rank factorization of a q-parameter polynomial m × n matrix F of full row rank (called the PG-q factorization) of the form F = PG, where is the greatest left divisor of F; Δ _i ^(k) i is a regular (q-k)-parameter polynomial matrix the characteristic polynomial of which is a primitive polynomial over the ring of polynomials in q-k-1 variables, and G is a q-parameter polynomial matrix of rank m. The PG-q algorithm is suggested, and its applications to solving some problems of algebra are presented. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 150–163.     

To solving multiparameter problems of algebra. 6. Spectral characteristics of polynomial matrices

For a q-parameter polynomial m × n matrix F of rank ρ, solutions of the equation Fx = 0 at points of the spectrum of the matrix F determined by the (q −1)-dimensional solutions of the system Z[F] = 0 are considered. Here, Z[F] is the polynomial vector whose components are all possible minors of order ρ of the matrix F. A classification of spectral pairs in terms of the matrix A[F], with which the vector Z[F] is associated, is suggested. For matrices F of full rank, a classification and properties of spectral pairs in terms of the so-called levels of heredity of points of the spectrum of F are also presented. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 132–149.     

Solution of systems of general-form nonilinear algebraic equations. Methods and algorithms. IV

This paper considers the solution of a system of m nonlinear equations in q>02 variables (SNAE-q). A method for finding all of the finite zero-dimensional roots of a given SNAE-q, which extends the method suggested previously for q=2 and q=3 to the case q≥2, is developed and theoretically justified. This method is based on the algorithm of the ΔW-q factorization of a polynomial q-parameter matrix and on the algorithm of relative factorization of a scalar polynomial in q variables. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 124–146. Translated by V. N. Kublanovskaya.     

To Solving Multiparameter Problems of Algebra. 5. The ∇V-q Factorization Algorithm and its Applications

The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 144–153.     

Irreducible factorizations of rationalq-parameter matrices

This paper considers rational q-parameter matrices (i.e., matrices the entries of which are ratios of scalar polynomials in q variables) and extends the previous results of the authors. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 147–164. Translated by V. N. Kublanovskaya.     

To solving multiparameter problems of algebra 3. Cylindrical manifolds of the regular spectrum of a matrix

Methods for computing polynomials (complete polynomials) whose zeros form cylindrical manifolds of the regular spectrum of a q-parameter polynomial matrix in the space ^q are considered. Based on the method of partial relative factorization of matrices, new methods for computing cylindrical manifolds are suggested. The W and V methods, previously proposed for computing complete polynomials of q-parameter polynomial matrices whose regular spectrum is independent of one of the parameters, are extended to a wider class of matrices. Bibliography: 4 titles._______Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 108–121.     

To solving problems of algebra for two-parameter matrices. I

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, μ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, μ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, μ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 107–149.     

Spectral problems for pencils of polynomial matrices. Methods and algorithms. V

Methods and algorithms for the solution of spectral problems of singular and regular pencils D(λ, μ)=A(μ)-λB(μ) of polynomial matrices A(μ) and B(μ) are suggested (the separation of continuous and discrete spectra, the computation of points of a discrete spectrum with the corresponding, Jordan chains, the computation of minimal indices and a minimal basis of polynomial solutions, the computation of the determinant of a regular pencil). Bibliography: 13 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 26–70     

To solving multiparameter problems of algebra. 2. The method of partial relative factorization and its applications

For a q-parameter (q 2) polynomial matrix of full rank whose regular and singular spectra have no points in common, a method for computing its partial relative factorization into a product of two matrices with disjoint spectra is suggested. One of the factors is regular and is represented as a product of q matrices with disjoint spectra. The spectrum of each of the factors is independent of one of the parameters and forms in the space ^q a cylindrical manifold w.r.t. this parameter. The method is applied to computing zeros of the minimal polynomial with the corresponding eigenvectors. An application of the method to computing a specific basis of the null-space of polynomial solutions of the matrix is considered. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 89–107.     

To solving problems of algebra for two-parameter matrices. IV

This paper continues the series of publications devoted to surveying and developing methods for solving the following problems for a two-parameter matrix F (λ, μ) of general form: exhausting points of the mixed regular spectrum of F (λ, μ); performing operations on polynomials in two variables (computing the GCD and LCM of a few polynomials, division of polynomials, and factorization); computing a minimal basis of the null-space of polynomial solutions of the matrix F (λ, μ) and separation of its regular kernel; inversion and pseudo in version of polynomial and rational matrices in two variables, and solution of systems of nonlinear algebraic equations in two unknowns. Most of the methods suggested are based on rank factorizations of a two-parameter polynomial matrix and on the method of hereditary pencils. Bibliography: 8 titles.     

To solving problems of algebra for two-parameter matrices. III

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. It considers linearization methods, which allow one to reduce the problem of solving an equation F(λ, μ)x = 0 with a polynomial two-parameter matrix F(λ, μ) to solving an equation of the form D(λ, μ)y = 0, where D(λ, μ) = A(μ)-λB(μ) is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix F(λ, μ) are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains of vectors are developed. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 166–207.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

To solving spectral problems for <Emphasis Type="Italic">q</Emphasis>-parameter polynomial matrices

The method of hereditary pencils, originally suggested by the author for solving spectral problems for two-parameter matrices (pencils of matrices), is extended to the case of q-parameter, q ≥ 2, polynomial matrices. Algorithms for computing points of the finite regular and singular spectra of a q-parameter polynomial matrix and their theoretical justification are presented. Bibliography: 2 titles.     

Operations over scalar polynomials and their computer realization

In this paper, algorithms which realize some operations over scalar polynomials in one and two variables and their computer realization are suggested. The following operations are considered: 1) the computation of the GCD for given scalar polynomials and the decomposition of each polynomial into a product of two factors: the first factor is the GCD, and the second factors form a sequence of relatively prime polynomials; 2) the division of polynomials by their common divisor; 3) the decomposition of polynomials in two variables into irreducible factors; 4) the computation of the LCM for given scalar polynomials. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 158–175. This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. N. Kublanovskaya.     

Divisibility of polynomials over finite fields and combinatorial applications

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252–260, 1998) uses trinomials over \mathbbF₂{\mathbb{F}_2} to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1–17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over \mathbbF₂{\mathbb{F}_2} . Here we first simplify the requirement in Munemasa’s approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over \mathbbF₃{\mathbb{F}_3} . Some of our results apply to any finite field \mathbbF_q{\mathbb{F}_q} with q elements.     

Differential operators associated with zonal polynomials. II

Summary LetC _κ(S) be the zonal polynomial of the symmetricm×m matrixS=(s_ij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.