Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables

TL;DR

This paper derives an explicit formula for counting ideals of a fixed codimension in the algebra of Laurent polynomials in two variables over a finite field, revealing palindromic polynomial structures and connections to divisor sums.

Contribution

It provides a new explicit formula for the number of ideals of given codimension in Laurent polynomial algebras, linking algebraic counts to divisor sums and subgroup enumeration.

Findings

Number of ideals is a palindromic polynomial in q.

The count relates to a q-analogue of the sum of divisors of n.

The formula connects algebraic ideals to subgroup enumeration in Z^2.

Abstract

We establish an explicit formula for the number $C_n(q)$ of ideals of codimension $n$ of the algebra ${\mathbb F}_q[x,y,x^{-1}, y^{-1}]$ of Laurent polynomials in two variables over a finite field of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n(q) = (q-1)^2 P_n(q)$, where $P_n(q)$ 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 $n$.