The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 67, Issue 4 (2018), 715-741.
Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables
We establish an explicit formula for the number of ideals of codimension (colength) of the algebra of Laurent polynomials in two variables over a finite field of cardinality . This number is a palindromic polynomial of degree in . Moreover, , where is another palindromic polynomial; the latter is a -analogue of the sum of divisors of , which happens to be the number of subgroups of of index .
Michigan Math. J., Volume 67, Issue 4 (2018), 715-741.
Received: 23 November 2016
Revised: 2 February 2018
First available in Project Euclid: 16 June 2018
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 14C05: Parametrization (Chow and Hilbert schemes) 14N10: Enumerative problems (combinatorial problems) 16S34: Group rings [See also 20C05, 20C07], Laurent polynomial rings
Secondary: 05A30: $q$-calculus and related topics [See also 33Dxx] 11P84: Partition identities; identities of Rogers-Ramanujan type 11T55: Arithmetic theory of polynomial rings over finite fields 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14G15: Finite ground fields
Kassel, Christian; Reutenauer, Christophe. Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables. Michigan Math. J. 67 (2018), no. 4, 715--741. doi:10.1307/mmj/1529114453. https://projecteuclid.org/euclid.mmj/1529114453