On Regular and Logarithmic Solutions of Ordinary Linear Differential Systems

Higher-Order Linear Differential Systems with Truncated Coefficients

“…In the decidable cases, this means that we can reduce the problem of finding Laurent series solutions of systems with power series coefficients to that of finding the same type of solutions for systems with polynomial coefficients. A number of methods exist to do this task efficiently (e.g., [1,6]). The mathematical techniques we employ in this paper use the algebra of polynomials and matrices, and we give explicit formulae for finding a l for a given l. An implementation of our results can be done easily in any computer algebra system, as demonstrated in the previous section for the Maple package ISOLDE, and this equally applies to the implementations of the algorithms from [1,6].…”

“…A number of methods exist to do this task efficiently (e.g., [1,6]). The mathematical techniques we employ in this paper use the algebra of polynomials and matrices, and we give explicit formulae for finding a l for a given l. An implementation of our results can be done easily in any computer algebra system, as demonstrated in the previous section for the Maple package ISOLDE, and this equally applies to the implementations of the algorithms from [1,6]. We hope that this paper hence also makes a practical contribution to the scientific computing community, wishing to use computer algebra for handling systems of linear differential equations.…”

“…(ii) There exist systems of the form (1) for which the truncation problem has no solution (no sequence (a i ) exists); the algorithmic problem of recognizing whether or not the truncation problem for a given system of the form (1) has a solution (a sequence (a i )) is undecidable.…”

“…Laurent series solutions of such systems form a building block for other types of solutions, and more generally, algorithms for finding Laurent series solutions may be a part of various computer algebra algorithms (see e.g. [1,6]). …”

“…We denote by k [[x]] the ring of formal power series with coefficients in k and k ((x) Let a system S be of the form (1) and define the l-truncation S l which is obtained by omitting all the terms of degree larger than or equal to l in the coefficients of S.…”