Michigan Math. J. 67 (2018), 715--741; arXiv:1505.07229v4

Christian Kassel and Christophe Reutenauer

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

Mathematics Subject Classification (2000): Primary 05A17, 14C05, 14G10, 14N10, 16S34, Secondary 05A30, 11P84, 11T55, 13P10, 14G15

Abstract. We establish an explicit formula for the number C_n(q) of ideals of codimension n of the algebra 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 Z² of index n.

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(28 novembre 2018)