A relationship between the ideals of ${\mathbb{F}}_q\left[x,y,x^{-1},y^{-1}\right]$ and the Fibonacci numbers

Abstract

Let $C_n(q)$ be the number of ideals of codimension $n$ of ${\mathbb{F}}_q\left[x,y,x^{-1},y^{-1}\right]$, where ${\mathbb{F}}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial in $q$ for any fixed value of $n\ge 1$. For $q=\frac{3+\sqrt{5}}{2}$, this combinatorial interpretation of $C_n(q)$ is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears. Let $f_n$ be the $n$-th Fibonacci number (following the convention $f_0=0$, $f_1=1$). Define the series $$F(t)=\sum _{n\ge 1}{f}_{2n}{t}^n.$$ We will prove that for each $n\ge 1$, $$C_n\left(\frac{3+\sqrt{5}}{2}\right)={\lambda }_n\left(f_{2n}\frac{3+\sqrt{5}}{2}-f_{2n-2}\right),$$ where the integers ${\lambda }_n\ge 0$ are given by the following generating function $$\prod _{m\ge 1}\left(1+F\left(t^m\right)\right)=1+\sum _{n\ge 1}{\lambda }_n{t}^n.$$

I do not agree anymore with the ideas expressed in the manuscript

References in corpus (1)

• Complete determination of the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus