Telescoping Algorithms for $Sigma^*$-Extensions via Complete Reductions

2025-06-10

AI Summary

This paper addresses the parameterized telescoping summation problem within Σ*-extension towers containing harmonic numbers and other iterated sums. Method: Leveraging difference field theory and linear operator decomposition, we construct a decidable complete reduction operator that uniquely decomposes any element into a summable part and a zero remainder. Contributions/Results: (1) We establish, for the first time in Σ*-extension towers, an equivalence between summability and vanishing remainder—a decidable characterization of telescoping; (2) we uncover a deep structural connection to Karr’s fundamental theorem; and (3) we significantly improve telescoping algorithm efficiency, enabling automatic depth reduction of nested sums. Our framework provides both a theoretical foundation and a practical tool for structured decidability and automated simplification of complex multi-layered sums arising in combinatorics, particle physics, and related domains.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Natural Language Processing: Summarization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is equal to zero. In this paper, we present a complete reduction in a tower of $Sigma^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $Sigma^*$-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.

Problem

• Develop complete reduction in Σ*-extensions for telescoping
• Efficiently solve parameterized telescoping problem in sums
• Connect complete reductions to Karr's Fundamental Theorem

Innovation

• Complete reduction in Σ*-extensions tower
• Efficient framework for telescoping problem
• Reduces sum depth with structural theorems