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**Computations with Frobenius powers.**
*(English)*
Zbl 1101.13006

Summary: It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. M. Katzman [J. Algebra 203, No. 1, 211–225 (1998; Zbl 0919.13007)] 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 Gröbner bases in reverse lexicographic ordering of the ideals of the form \(J + \mathbb{F}_qI\). 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 Gröbner bases connected with Katzman’s conjectures.

### MSC:

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |