  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.

Commutative Algebra