Some well-known correspondences between sets of linearly independent rows and columns of matrices over fields carry over to matrices over non-commutative rings without nontrivial zero divisors.

Algorithms for computing lower bounds on valuations (e.g., orders of the poles) of the components of meromorphic solutions of arbitrary-order linear difference systems with polynomial coefficients are considered. In addition to algorithms based on ideas which have been already utilized in computer algebra for treating normal first-order systems, a new algorithm using tropical calculations is proposed. It is shown that the latter algorithm is rather fast, and produces the bounds with good accuracy.

We consider linear q-difference equations with polynomial coefficients depending on parameters. For the case when the ground field is Q(q) we propose an algorithm recognizing whether or not there exist numerical values of parameters for which a given equation has a non-zero polynomial solution (alternatively, a rational-function solution). We prove that there exists no such algorithm if the parameter values are polynomials or rational functions of q.

No abstract available.

We consider sequences which satisfy a linear recurrence equation Ly = 0 with polynomial coefficients. A criterion, i.e., a necessary and sufficient condition is proposed for validity of the discrete Newton-Leibniz formula when a primitive (an indefinite sum) Rt of a solution t of Ly = 0 is obtained either by Gosper's algorithm or by the Accurate Summation algorithm (the operator R has rational-function coefficients, ordR = ordL−1; in the Gosper case ordL = 1, ordR = 0). Additionally we show that if Gosper's algorithm succeeds on L, ordL = 1, then Ly = 0 always has some nonzero solutions t, defined everywhere, such that the discrete Newton-Leibniz formula Ewk=v t(k) = u(w+1)−u(v) is valid for u = Rt and any integer bounds v≤w.

Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper's algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.

We propose four multiplicative canonical forms that exhibit the shift structure of a given rational function. These forms in particular allow one to represent a hypergeometric term efficiently. Each of these representations is optimal in some sense.

A terminating condition of the well-known Zeilberger's algorithm for a given hypergeometric term T(n, k) is presented. It is shown that the only information on T(n, k) that one needs in order to determine in advance whether this algorithm will succeed is the rational function T(n, k + 1)/T(n, k).

We describe a new direct algorithm for transforming a linear system of recurrences into an equivalent one with nonsingular leading or trailing matrix. Our algorithm, which is an improvement to the EG elimination method [2], uses only elementary linear algebra operations (ranks, kernels and determinants) to produce an equation satisfied by the degrees of the solutions with finite support. As a consequence, we can bound and compute the polynomial and rational solutions of very general linear functional systems such as systems of differential or (q—) difference equations.

We present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) = T1(n + 1) -T1(n) + T2(n), and T2(n) is minimal in some sense. This solves the decomposition problem for indefinite sums of hypergeometric terms: T1(n + 1) - T1(n) is the “summable part” and T2(n) the “non-summable part” of T(n).

We describe an algorithm for finding the positive integer solutions n of orbit problems of the form αn = β where α and β are given elements of a field K. Our algorithm corrects the bounds given in [7], and shows that the problem is not polynomial in the Euclidean norms of the polynomials involved. Combined with a simplified version of the algorithm of [8] for the “specification of equivalence”, this yields a complete algorithm for computing the dispersion of polynomials in nested hypergeometric extensions of rational function fields. This is a necessary step in computing symbolic sums, or solving difference equations, with coefficients in such fields. We also solve the related equations p(αn) = 0 and p(n, αn) = 0 where p is a given polynomial and α is given.

We introduce the notion of the adjoint Ore ring and give a definition ofadjoint polynomial, operator andequation.We apply thk for integrating solutions of Ore equations.

We propose an algorithm to construct the minimal annihilating operator of a function or a sequence, when the operator is completely factorable (i.e. can be decomposed in first order factors).The algorithm is designed in the frame of the Ore rings theory and can be used in the differentiaf, difference and q-difference cases.We describe also a Maple implement ation of the algorithm.1 Introduction Constructing a linear ordinary differential operator annihilating a function (an annihilator of the function) is necessary when solving many computer algebra problems.We list some of these problems.P1. Expanding a function as a power series and subsequently investigating the expansion.An annihilator lets one construct the recurrence for the series coefficients and manipulate them ([14, 17]). P2.Solving linear inhomogeneous equations.Some methods use annihilators of the right-hand side ([4, 8]).P3. Integrating.If the minimal annihilator L, ord L = n, of j is given, then one can check whether there exists a primitive of ~with an n-th order minimal annihilator.If yes, then it is possible to express the primitive explicitly via f ([91) P4.Recognizing the equivalence of two given functions.If the common annihilator of both the functions is given, then it suffices to check the agreement between the corresponding "initial conditions" (a classical approach).The minimal annihilator, i.e. the annihilator of the lowest order, is the most informative.Note that to solve P3 only the minimal annihilator of j is suitable.Applying algorithm [8] to an equation with a d'Alembertian righthand side guarantees that all d'Alembertian solutions will be found only in the situation when the minimaf annihilator of the right hand side, decomposed in first order factors, is given.(A function is d'Alembertian if it has a completely -Work reported herein was supported in part by RFBR under Grant 95-01-01138.

Let an Ore polynomial ring k[X; a, 6] and a nonzero pseudolinear map 19: K + K, where K is a O, &compatible extension of the field k, be given.Then we have the ring k[O] of op-

Article Free Access Share on Rational solutions of linear difference and q-difference equations with polynomial coefficients Author: S. A. Abramov Computer Center of the Russian Academy of Science, Vavilova 40, Moscow 117967, Russia Computer Center of the Russian Academy of Science, Vavilova 40, Moscow 117967, RussiaView Profile Authors Info & Claims ISSAC '95: Proceedings of the 1995 international symposium on Symbolic and algebraic computationApril 1995 Pages 285–289https://doi.org/10.1145/220346.220383Published:01 April 1995Publication History 29citation422DownloadsMetricsTotal Citations29Total Downloads422Last 12 Months34Last 6 weeks13 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We propose a new algorithm for indefinite rational summation which, given a rational function F(z), extracts a rational part R(z) from the indefinite sum of F(z): @1995

Introduction Let K be a field of characteristic 0 and L : K[x] ! K[x] an endomorphism of the K-linear space of univariate polynomials over K. We consider the following computational tasks concerning L: T1. Homogeneous equation Ly = 0: Compute a basis of Ker L in K[x]. T2. Inhomogeneous equation Ly = f : Given f 2 K[x], compute a basis of the affine space L \\Gamma1 (f) in K[x]. T3. Parametric inhomogeneous equation Ly = P m i=1 i f i : Given f1

D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reducing the order. Knowing d'Alembertian solutions of Ly = 0, one can write down the corresponding solutions of Ly = f and of L*y = 0.

Article Free Access Share on On d'Alembert substitution Author: S. A. Abramov View Profile Authors Info & Claims ISSAC '93: Proceedings of the 1993 international symposium on Symbolic and algebraic computationAugust 1993 Pages 20–26https://doi.org/10.1145/164081.164086Published:01 August 1993Publication History 6citation227DownloadsMetricsTotal Citations6Total Downloads227Last 12 Months12Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Article Free Access Share on On the greatest common divisor of polynomials which depend on a parameter Authors: S. A. Abramov View Profile , K. Yu. Kvashenko View Profile Authors Info & Claims ISSAC '93: Proceedings of the 1993 international symposium on Symbolic and algebraic computationAugust 1993 Pages 152–156https://doi.org/10.1145/164081.164112Published:01 August 1993Publication History 11citation235DownloadsMetricsTotal Citations11Total Downloads235Last 12 Months18Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Article Free Access Share on Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients Authors: S. A. Abramov Computer Center of the USSR Acad. of Sci. Computer Center of the USSR Acad. of Sci.View Profile , K. Yu. Kvansenko Computer Center of the USSR Acad. of Sci. Computer Center of the USSR Acad. of Sci.View Profile Authors Info & Claims ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computationJune 1991 Pages 267–270https://doi.org/10.1145/120694.120735Online:01 June 1991Publication History 28citation285DownloadsMetricsTotal Citations28Total Downloads285Last 12 Months4Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF