Complexity of powers of a constant-recursive sequence

TL;DR

This paper investigates the complexity of powers of constant-recursive sequences, especially when the roots of their characteristic polynomial are not all distinct, extending understanding of their algebraic structure.

Contribution

It provides new insights into the rank behavior of powers of constant-recursive sequences with repeated roots, answering a previously open question.

Findings

Analyzes the rank of sequence powers with non-distinct roots

Establishes bounds on the complexity of these sequences

Extends prior results to more general cases

Abstract

Constant-recursive sequences are those which satisfy a linear recurrence, so that later terms can be obtained as a linear combination of the previous ones. The rank of a constant-recursive sequence is the minimal number of previous terms required for such a recurrence. For a constant-recursive sequence $s(n)$, we study the sequence $\left(\text{rank}\, s(n)^M\right)_{M\geq 1}$. We answer a question of Stinchcombe regarding the complexity of the powers of a constant-recursive sequence when the roots of the characteristic polynomial are not all distinct.