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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.15780 |
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| _version_ | 1866908529517920256 |
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| author | Kassel, Christian Reutenauer, Christophe |
| author_facet | Kassel, Christian Reutenauer, Christophe |
| contents | 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 $C_n(q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pairs of intertwined integer sequences Kassel, Christian Reutenauer, Christophe Number Theory Combinatorics 11T55, 14N10 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 $C_n(q)$. |
| title | Pairs of intertwined integer sequences |
| topic | Number Theory Combinatorics 11T55, 14N10 |
| url | https://arxiv.org/abs/2507.15780 |