Pairs of intertwined integer sequences

TL;DR

This paper explores the properties of polynomials related to ideals in Laurent polynomial algebras, revealing their connections to Chebyshev polynomials and providing explicit formulas and conditions for their equality.

Contribution

It introduces a new polynomial sequence linked to ideal counts, expressing it via Chebyshev polynomials and characterizing when it equals a specific polynomial.

Findings

${ar{P}}_n(X)$ closely approximates $F_{n-1}(X)$ for all integers $N$

${ar{P}}_n(X)$ equals $F_{n-1}(X)$ if and only if $n$ is a power of 2

Explicit formulas for ${ar{P}}_n(X)$ in terms of Chebyshev polynomials

Abstract

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field: it turned out that $C_n(q)$ is a palindromic polynomial of degree $2n$ in $q$, divisible by $(q-1)^2$. The quotient $P_n(q) = C_n(q)/(q-1)^2$ is a palindromic polynomial of degree $2n-2$. For each $n\geq 1$ let ${\overline{P}}_n(X) \in {\mathbb{Z}}[X]$ be the degree $n-1$ polynomial such that ${\overline{P}}_n(q+q^{-1}) = P_n(q)/q^{n-1}$. In this note we show that for any integer $N$ the integer value ${\overline{P}}_n(N)$ is close to the value at $N$ of the degree $n-1$ polynomial $F_{n-1}(X) = 1 + \sum_{k=1}^{n-1} \, {\overline{T}}_k(X)$, which is a sum of monic versions ${\overline{T}}_k(X)$ of Chebyshev polynomials of the first kind. We give a precise formula for ${\overline{P}}_n(X)$ as a linear combination of $F_k(X)$'s, each appearance of the latter being parametrized by an odd divisor of $n$. As a consequence, ${\overline{P}}_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of $2$. We exhibit similar formulas for $C_n(q)$.