Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables

Abstract

We establish an explicit formula for the number $C_n\left(q\right)$ of ideals of codimension (colength) $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1},y^{-1}]$ of Laurent polynomials in two variables over a finite field ${\mathbb{F}}_q$ of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n\left(q\right)=(q-1{)}^2{P}_n(q)$, where $P_n\left(q\right)$ is another palindromic polynomial; the latter is a $q$-analogue of the sum of divisors of $n$, which happens to be the number of subgroups of ${\mathbb{Z}}^2$ of index $n$.