Computing with D-Algebraic Sequences
Bertrand Teguia Tabuguia
TL;DR
The paper formalizes D-algebraic sequences as zeros of algebraic difference equations ($ADE$s) over a characteristic-zero algebraically closed field and develops a constructive calculus for them. It introduces a robust framework based on difference rings, defines generic zeros and regularity, and proves closure properties under arithmetic operations via a decomposition-elimination-prolongation approach. It then shows that holonomic and $C^2$-finite sequences are D-algebraic by converting their defining relations into rationalizing $ADE$s, providing algorithms and concrete examples, and establishes that subsequences along arithmetic progressions remain D-algebraic with order scaling by the progression step. Together, these results furnish a comprehensive symbolic toolkit for manipulating nonlinear, difference-algebraic sequences with potential applications in acceleration, modeling, and computation.
Abstract
A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations are denoted algebraic difference equations (ADEs). We propose a formal definition of D-algebraicity for sequences and investigate algorithms for their closure properties. We show that subsequences of D-algebraic sequences, indexed by arithmetic progressions, satisfy ADEs of the same orders as the original sequences. Additionally, we discuss the special difference-algebraic nature of holonomic and $C^2$-finite sequences.