Complete reduction method speeds up telescoping in complex sum models
Complete Reductions and Idempotent Representations for $RΠΣ^*$-towers
Symbolic Computation
Summary
The paper deals with a complex math problem of breaking down very complicated sums and products into simpler pieces, a common challenge in combinatorics and physics. The authors present a new way to split these sums without solving difficult equations, making it easier to decide if a sum can be simplified or not. Their approach speeds up calculations that involve finding patterns or recurrences in sums, which can be useful whenever nested sums or products appear. They also extend their method to cover even broader cases that were not manageable before.
What this means in practice
- •For symbolic computation developers: Implement faster algorithms for simplifying and solving nested sums and products that appear in combinatorial and algebraic computations.
- •For mathematical physicists: Use the new reduction and decomposition methods to handle complicated sums in particle physics calculations more efficiently.
Authors
Yiman Gao, Jakob Obrovsky, Carsten Schneider
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.