Computing generalized Frobenius powers of monomial ideals

TL;DR

This paper introduces an algorithm based on linear optimization to compute critical exponents of monomial ideals, advancing understanding of their Frobenius powers and critical exponents in characteristic-dependent contexts.

Contribution

The paper presents a novel algorithm for calculating critical exponents of monomial ideals, providing new insights into their Frobenius powers without relying on test ideals.

Findings

Algorithm effectively computes critical exponents for monomial ideals.

Results include new bounds and properties of Frobenius powers.

Demonstrates the utility of linear optimization techniques in algebraic computations.

Abstract

Generalized Frobenius powers of an ideal were introduced in work of Hern\'andez, Teixeira, and Witt as characteristic-dependent analogs of test ideals. However, little is known about the Frobenius powers and critical exponents of specific ideals, even in the monomial case. We describe an algorithm to compute the critical exponents of monomial ideals and use this algorithm to prove some results about their Frobenius powers and critical exponents. Rather than using test ideals, our algorithm uses techniques from linear optimization.