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

Abstract

We establish an explicit formula for the number Cn(q) of ideals of codimension (colength) n of the algebra Fq[x,y,x−1,y−1] of Laurent polynomials in two variables over a finite field Fq of cardinality q. This number is a palindromic polynomial of degree 2n in q. Moreover, Cn(q)=(q−1)2Pn(q), where Pn(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 Z2 of index n.