Complete Reductions and Idempotent Representations for RΠΣ^*-towers

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.

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BibTeX RIS

BibTeX

@misc{gao2026complete,
  title = {{Complete Reductions and Idempotent Representations for $RΠΣ^*$-towers}},
  author = {Yiman Gao and Jakob Obrovsky and Carsten Schneider},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24845},
  eprint = {2609.24845},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24845},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Complete Reductions and Idempotent Representations for RΠΣ^*-towers
AU  - Yiman Gao
AU  - Jakob Obrovsky
AU  - Carsten Schneider
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24845
UR  - https://arxiv.org/abs/2609.24845
AB  - 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.
ER  -

Abstract

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