Dynamical sequences: closure properties and automatic identity proving

TL;DR

The paper introduces dynamical sequences as $a(n)=f(\varphi^n(x_0))$ from algebraic dynamics over an algebraically closed field $K$, and develops a comprehensive framework of closure properties, notable contained families (including all $C^n$- and $D^n$-finite sequences, Somos sequences, elliptic divisibility sequences, and subsequences of linear recurrences), and an algorithm for automatic identity proving. It shows the class forms a $K$-algebra under pointwise operations and demonstrates breadth via explicit constructions and examples, including $D^n$-finite and $C^n$-finite sequences, Somos and EDS, and subsequences such as $f(d^n)$ and $f(P(n))$. The identity-checking algorithm reduces equality questions to finitary checking of Zariski-closed sets, with concrete SageMath/Groebner-basis verifications for several classical identities. The work opens several directions, including convolution-closure, Skolem-type decidability, and the structure of zero sets and height growth in dynamical sequences, connecting algebraic dynamics with computational identity verification.

Abstract

Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{A}^1$ are rational maps defined over $K$, and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the class of dynamical sequences enjoys numerous closure properties and encompasses all elliptic divisibility sequences, all Somos sequences, and all $C^n$- and $D^n$-finite sequences for all $n\ge 1$, as defined by Jiménez-Pastor, Nuspl, and Pillwein. We also give an algorithm for proving that two dynamical sequences are identical and illustrate how to use this algorithm by showing how to prove several classical combinatorial identities via this method.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['thm:main']}
• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof