发表机构
Johannes Kepler University Linz（林茨约翰·开普勒大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对$R\Pi\Sigma^*$-扩张提出完全归约方法，通过构造差分补空间分解元素，无需解差分方程即可判定可求和性，并给出幂等表示以推广伸缩算法，显著加速参数化与创造性伸缩。
AI 中文摘要
$R\Pi\Sigma^*$-扩张构成了一类丰富的差分环，为建模不定嵌套和、超越乘积以及根上单位结构上的嵌套乘积提供了统一的代数框架，这些结构频繁出现在组合学、数论和粒子物理中。对于这些扩张中常数环为域的一大子类，我们引入了一种完全归约方法来解决伸缩问题，而无需求解任何差分方程。更精确地说，我们显式地构造了常数域上差分子空间的一个补空间，并开发了一种算法，将扩张中的任意元素分解为一个差分与一个位于该补空间中的分量之和。因此，可求和性当且仅当该补分量等于零时成立。这种结构化方法显著加速了参数化伸缩，尤其是创造性的伸缩，用于推导定和线性递推。最后，我们计算了一个显式的幂等表示，将现有的伸缩算法和我们的完全归约框架推广到一般的$R\Pi\Sigma^*$-扩张类，开辟了以前无法处理的类和乘积。
英文摘要
$RΠΣ^*$-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.
