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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 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.
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Contributor : Christian Kassel <>
Submitted on : Thursday, June 25, 2020 - 11:35:49 AM
Last modification on : Friday, June 26, 2020 - 3:33:36 AM

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Christian Kassel, Christophe Reutenauer. Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables. The Michigan Mathematical Journal, Michigan Mathematical Journal, 2018, 67 (4), pp.715-741. ⟨10.1307/mmj/1529114453⟩. ⟨hal-02880825⟩

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