The Michigan Mathematical Journal

Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables

Christian Kassel and Christophe Reutenauer

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Abstract

We establish an explicit formula for the number Cn(q) of ideals of codimension (colength) n of the algebra Fq[x,y,x1,y1] of Laurent polynomials in two variables over a finite field Fq of cardinality q. This number is a palindromic polynomial of degree 2n in q. Moreover, Cn(q)=(q1)2Pn(q), where Pn(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.

Article information

Source
Michigan Math. J., Volume 67, Issue 4 (2018), 715-741.

Dates
Received: 23 November 2016
Revised: 2 February 2018
First available in Project Euclid: 16 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1529114453

Digital Object Identifier
doi:10.1307/mmj/1529114453

Mathematical Reviews number (MathSciNet)
MR3877434

Zentralblatt MATH identifier
07056366

Subjects
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

Citation

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


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