Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables (Q2414684)

Let \(\mathbb{F}_q\) be the finite field of cardinal \(q\) and \(A=\mathbb{F}_q[x,y,x^{-1},y^{-1}]\) be the algebra of Laurent polynomials in two variables \(x\) and \(y\) over \(\mathbb{F}_q\). The codimension of an ideal \(I\) of \(A\) is defined to be the dimension of the \(\mathbb{F}_q-\)vector space \(A/I\). In this paper, the authors establish an explicit formula for the number \(C_n(q)\) of ideals in \(A\) of a given codimension \(n\). This number is a particular polynomial of degree \(2n\) in the variable \(q\) and is related to some arithmetical functions.