Vol. 275, No. 2, 2015

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 314: 1
Vol. 313: 1  2
Vol. 312: 1  2
Vol. 311: 1  2
Vol. 310: 1  2
Vol. 309: 1  2
Vol. 308: 1  2
Vol. 307: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Explicit Hilbert–Kunz functions of $2 \times 2$ determinantal rings

Marcus Robinson and Irena Swanson

Vol. 275 (2015), No. 2, 433–442
Abstract

Let k[X] = k[xi,j : i = 1, , m; j = 1, , n] be the polynomial ring in mn variables xi,j over a field k of arbitrary characteristic. Denote by I2(X) the ideal generated by the 2 × 2 minors of the generic m × n matrix [xi,j]. We give a closed polynomial formulation for the dimensions of the k-vector space k[X](I2(X) + (x1,1q, , xm,nq)) as q varies over all positive integers, i.e., we give a closed polynomial form for the generalized Hilbert–Kunz function of the determinantal ring k[X]I2(X). We also give a closed formulation of dimensions of other related quotients of k[X]I2(X). In the process we establish a formula for the numbers of some compositions (ordered partitions of integers), and we give a proof of a new binomial identity.

Keywords
Hilbert–Kunz function, multiplicity, combinatorial identity
Mathematical Subject Classification 2010
Primary: 13D40
Secondary: 05A15, 05A10
Milestones
Received: 4 March 2014
Revised: 16 December 2014
Accepted: 17 December 2014
Published: 15 May 2015
Authors
Marcus Robinson
Department of Mathematics
University of Utah
155 S 1400 E
John Widtsoe Building 233
Salt Lake City, UT 84112
United States
Irena Swanson
Department of Mathematics
Reed College
3203 SE Woodstock Blvd
Portland, OR 97202-8199
United States