Computations with Frobenius powers
Abstract
It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. Katzman showed that tight closure of ideals in these rings commutes with localization at one element if for all ideals I and J in a polynomial ring there is a linear upper bound in q on the degree in the least variable of reduced Groebner bases in reverse lexicographic ordering of the ideals of the form J + I^{[q]}. Katzman conjectured that this property would always be satisfied. In this paper we prove several cases of Katzman's conjecture. We also provide an experimental analysis (with proofs) of asymptotic properties of Groebner bases connected with Katzman's conjectures.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2002
 arXiv:
 arXiv:math/0212238
 Bibcode:
 2002math.....12238H
 Keywords:

 Mathematics  Commutative Algebra;
 13P10;
 13A35
 EPrint:
 21 pages, incorporated changes in Section 2, due to discussions with Aldo Conca and Enrico Sbarra