Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.

We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.

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 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.

In this paper we consider deterministic computation of the exact determinant of a dense matrix M of integers. We present a new algorithm with worst case complexity O \\Gamma n 4 (log n + log jjM jj) + n 3 log 2 jjM jj \\Delta , where n is the dimension of the matrix and jjM jj is a bound on the entries in M , but with average expected complexity O \\Gamma n 4 + n 3 (log n + log jjM jj) 2 \\Delta , assuming some plausible properties about the distribution of M . We will also describe a practical version of the algorithm and include timing data to compare this algorithm with existing ones. Our result does not depend on "fast" integer or matrix techniques. 1 Introduction One of the most fundamental characteristics of a square matrix is its determinant. Its being 0 expresses the non-- invertibility of the matrix, i.e. the fact that it has a non-- trivial kernel. For a real matrix, its absolute value is the volume of the multi--dimensional parallelepiped with generating ed...

We describe a new algorithm for computing the solutions in F = C(& ,s f(,ckiJ: ) of linear ordinary differential equations with coefficients in F. Colnpared to the gcncral algorithm of [9], our algorithm avoids the computation of exponential solutions of equations wit,11 coefficients in C(z), its well as the solving of linear differential systems over C(z).Our mct,hod is effective and has been implemented.

We present alternative algorithms for computing symmetric powers of linear ordinary differential operators.Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains.The basic ideaa are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators.

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

We describe an efficient algorithm for computing the associated equations appearing in the Beke-Schlesinger factorisation method for linear ordinary differential operators. This algorithm, which is based on elementary operations with sets of integers, can be easily implemented for operators of any order, produces several possible associated equations, of which only the simplest can be selected for solving, and often avoids the degenerate case, where the order of the associated equation is less than in the generic case. We conclude with some fast heuristics that can produce some factorisations while using only linear computations.

We describe a rational algorithm that computes the full partial fraction expansion of a rational function over the algebraic closure of its field of definition. The algorithm uses only gcd operations over the initial field but the resulting decomposition is expressed with linear denominators. We give examples from its Axiom and Maple implementations. Introduction The partial fraction decomposition of a rational function is a form where both the local and global behaviour of the function are easy to find. This is used when computing a primitive by hand, or any linear operation which is most easily done on a pole. An example is the efficient computation of asymptotic expansion of the solutions of a linear recurrence with constant coefficients [4]. Let f = A=D be a rational function in some field K(z). By the fundamental theorem of algebra, it is clear that f admits a partial fraction decomposition of the form f = P + X D(ff)=0 n ff X i=1 b ff;i (z \\Gamma ff) i ; (1) where P is...

Article Linear ordinary differential equations: breaking through the order 2 barrier Share on Author: Manuel Bronstein View Profile Authors Info & Claims ISSAC '92: Papers from the international symposium on Symbolic and algebraic computationAugust 1992 Pages 42–48https://doi.org/10.1145/143242.143264Published:01 August 1992 18citation434DownloadsMetricsTotal Citations18Total Downloads434Last 12 Months9Last 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 SiteGet Access

Article The Risch differential equation on an algebraic curve Share on Author: Manuel Bronstein Informatik, ETH - Zentrum, CH-8092 Zürich, Switzerland Informatik, ETH - Zentrum, CH-8092 Zürich, SwitzerlandView Profile Authors Info & Claims ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computationJune 1991 Pages 241–246https://doi.org/10.1145/120694.120731Published:01 June 1991 2citation287DownloadsMetricsTotal Citations2Total Downloads287Last 12 Months5Last 6 weeks2 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 SiteGet Access

We describe an algorithm, based on Risch's real structure theorem, that determines explicitly all the algebraic relations among a given set of real elementary functions. We also provide examples from its implementation that illustrate the advantages over the use of complex logarithms and exponentials.