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International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 155-162
http://dx.doi.org/10.1155/S0161171291000157

On Hilbert polynomial of certain determinantal ideals

Department of Mathematics, S.P. College, Pune 411030, India

Received 2 February 1989; Revised 10 July 1989

Copyright © 1991 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X=(Xij) be an m(1) by m(2) matrix whose entries Xij, 1im(1), 1jm(2); are indeterminates over a field K. Let K[X] be the polynomial ring in these m(1)m(2) variables over K. A part of the second fundamental theorem of Invariant Theory says that the ideal I[p+1] in K[X], generated by (p+1) by (p+1) minors of X is prime. More generally in [1], Abhyankar defines an ideal I[p+a] in K[X], generated by different size minors of X and not only proves its primeness but also calculates the Hilbert function as well as the Hilbert polynomial of this ideal. The said Hilbert polynomial is completely determined by certain integer valued functions FD(m,p,a). In this paper we prove some important properties of these integer valued functions.