Factoring Finite-rank Linear Functional Systems∗

By a finite-rank (linear functional) system, we mean a system of linear differential, shift and q-shift operators or, any mixture thereof, whose solution space is finite-dimensional. This poster attempts to develop an algorithm for factoring finite-rank systems into “subsystems” whose solution spaces are of lower dimension. This work consists of three parts: 1. using module-theoretic language to interpret the factorization problem as that of finding submodules of so-called ∂-finite modules; 2. reducing the problem of finding submodules to that of finding onedimensional submodules of exterior products of ∂-finite modules; 3. computing hyperexponential solutions of (linear functional) matrix systems to find all the one-dimensional submodules. The first part is a generalization of the relevant material in Chapter 2 of [6]. The key ingredient of the second part is a generalization of Lemma 10 in [8] or arguments in §4.2.1 of [6], which connects d-dimensional submodules of a ∂-finite M and one-dimensional submodules of ∧M . The third part is essentially a direct application of algorithms described in [2, 3, 5]. It should be noted that the first algorithm for factoring finite-rank partial differential systems was developed in [4, 5]. Our work is motivated by this algorithm. Although their approach is ideal-theoretic and ours is module-theoretic, these two approaches have the same origin—the associated equations method for factoring linear ode’s by Beke and Schlesinger [1, 7]. Let A be an orthogonal Ore ring, a notion proposed recently in [2] to abstract common properties of linear differential, shift and q-shift operators. For convenience, we concern ourselves with the factorization problem of a finite-rank ideal (S) of A instead of a finite-rank system S. We can extend the interpretation of the factorization problem of linear ode’s as that of finding submodules of ordinary differential modules to the case of finite-rank ideals of A. Let I be a finite-rank ideal of A. ∗This poster reports joint work with M. Bronstein and Z. Li.