Solving linear difference equations with coefficients in rings with idempotent representations

TL;DR

This paper presents a reduction strategy for solving parameterized linear difference equations in difference rings by decomposing the ring into integral domains, enabling the use of existing solvers for complex algebraic structures.

Contribution

It introduces a novel reduction method that simplifies solving linear difference equations in complex difference rings with idempotent decompositions, extending solver applicability.

Findings

Reduction to solving in multiple integral domains

Applicable to rings with nested sums and products

Enables use of existing solvers in complex difference rings

Abstract

We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains. Utilizing existing solvers in this integral domain setting, we obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are built e.g., by $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields. This class of difference rings contains, e.g., nested sums and products, products over roots of unity and nested sums defined over such objects.