In this article, we consider the problem of computing polynomial solutions of general linear systems of ordinary integro-differential equations with polynomial coefficients. This algorithmic problem is a key step for many computations with matrices having linear integro-differential operator entries such as the computation of left/right syzygies, left/right inverses, left/right factorizations, and thus, for the development of an effective algebraic analysis approach for linear systems of ordinary integro-differential systems using effective elimination methods and effective homological algebra. The linear systems that appear in the above problems are generally rectangular and inhomogeneous. The contribution of this paper is to provide the first algorithm for computing polynomial solutions of inhomogeneous rectangular systems of linear integro-differential equations with polynomial coefficients. Our algorithm is implemented in the freely available Maple package Bavula.

Bavula proved that the ring \({\mathbb {I}}_1\) of polynomial ordinary integro-differential operators over a field \(\mathbb {k}\) of characteristic zero is coherent in the sense that the left/right kernel of any rectangular matrix with entries in \({\mathbb {I}}_1\) is a finitely generated left/right \({\mathbb {I}}_1\)-module. Unfortunately, his proof is not algorithmic. The contribution of this paper is to give an algorithmic proof of the coherence property of \({\mathbb {I}}_1\). We show that the kernel computation can be reduced to a kernel computation in a certain ring of skew Laurent polynomials and the computation of polynomial solutions of linear polynomial integro-differential systems. These two problems are shown to be effective. The algorithmic proof of the coherence of \({\mathbb {I}}_1\) allows us to develop an algorithmic elimination theory for linear systems of polynomial integro-differential equations with separable polynomial kernels. Finally, the algorithms presented in the paper are implemented in the freely available Maple package Bavula.

This paper provides a step forward to developing an algorithmic study of linear systems of polynomial ordinary integro-differential equations over a field <?TeX $\mathbb {k}$?> Math 1 of characteristic zero. Such a study can be achieved by first obtaining a constructive proof of the coherence property of the ring <?TeX ${\mathbb {I}}_1(\mathbb {k})$?> Math 2 of linear ordinary integro-differential operators with coefficients in <?TeX $\mathbb {k}[t]$?> Math 3 . To do that, the finiteness of the intersection of two finitely generated ideals has to be algorithmically studied. Three cases must be considered: first when evaluation operators generate the two ideals; second, when only one ideal is generated by evaluation operators; and third, when none is generated by evaluation operators. In this paper, we first explicitly characterize the intersection of two finitely generated ideals defined by evaluation operators. As for the second case, a key result is that the ideals generated by evaluations are semisimple <?TeX ${\mathbb {I}}_1$?> Math 4 -modules. We develop an algorithmic proof of this result. In particular, we show how a finite set of generators, defined by “simple” evaluations, can be obtained, that characterizes the class of finitely generated evaluation ideals of <?TeX ${\mathbb {I}}_1$?> Math 5 as finitely generated <?TeX $\mathbb {k}[t]$?> Math 6 -modules. Due to lack of space, the second and third cases will be developed in other publications.

This paper exposes some effective aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. More precisely, we prove that the annihilator of an evaluation operator is a finitely generated ideal which can be explicitly characterized and computed. This is an advance towards the development of an effective elimination theory for ordinary integro-differential operators and an effective study of linear systems of integro-differential equations with polynomial coefficients.