Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables
Résumé
We establish an explicit formula for the number C_n(q) of ideals of codimension n of the algebra 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)^2P_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 Z^2 of index n.