Complete Reductions and Idempotent Representations for $RΠΣ^*$-towers

2026-09-21

AI Summary

本文提出了一种完全约简方法，用于解决$R\Pi\Sigma^*$-扩展中的求和问题，无需解差分方程，并通过构造补空间及算法分解元素，加速了参数化与创造性求和过程。

Abstract

$RΠΣ^*$-extensions form a rich class of difference rings that provide a unified algebraic framework for modeling indefinite nested sums, transcendental products, and nested products over roots of unity structures that frequently appear in combinatorics, number theory, and particle physics. For a large subclass of these extensions whose ring of constants is a field, we introduce a complete reduction approach to resolve the telescoping problem without solving any difference equations. More precisely, we explicitly construct a complement to the subspace of differences over the constant field and develop an algorithm that decomposes any element of the extension into the sum of a difference and a component lying in this complement. Consequently, summability holds if and only if this complementary component is zero. This structural approach yields significant speed-ups for parameterized telescoping and, notably, creative telescoping for deriving linear recurrences of definite sums. Finally, we compute an explicit idempotent representation that extends existing telescoping algorithms and our complete reduction framework to the general class of $RΠΣ^*$-extensions, opening up previously untreatable classes of sums and products.

Problem

• difference rings
• telescoping problem
• sums and products

Innovation

• complete reduction
• idempotent representation
• $R\Pi\Sigma^*$-extensions
• telescoping problem
• linear recurrences