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

Abstract

We establish an explicit formula for the number $C_n(q)$ 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 $n$.

Section 1.4 and several references were added to Version 2. Version 3: a few lines added in Section 1.4. Version 4 (now 23 pages) is a shortened version: the number theory aspects of the paper were put into a separate publication (see arXiv:1610.07793)

References in corpus (1)

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