Symbolic powers and integral closures via extremal ideals

TL;DR

This paper introduces the use of extremal ideals to efficiently compute integral and symbolic powers of square-free monomial ideals, providing sharp bounds for various algebraic invariants and reducing complex algebraic problems to discrete geometry and linear programming.

Contribution

It demonstrates that extremal ideals can be used to compute integral closures and symbolic powers, offering sharp bounds and simplifying algebraic computations to geometric and linear programming problems.

Findings

Extremal ideals effectively compute integral and symbolic powers.

Sharp bounds are established for invariants like resurgence and Betti numbers.

Computations reduce to discrete geometry and linear programming.

Abstract

This paper demonstrates that extremal ideals can be used to great effect to compute integral closures of powers and symbolic powers of square-free monomial ideals. We show that the generators of these powers are images of the generators of the corresponding powers of extremal ideals under a specific ring homomorphism. Extremal ideals provide sharp bounds for a variety of invariants widely studied in the literature, including resurgence, asymptotic resurgence, and symbolic defect, as well as Betti numbers of symbolic powers and of integral closures of powers of square-free monomial ideals. When restricted to the class of extremal ideals, algebraic computations are reduced to problems of discrete geometry and linear programming, allowing the use of a wide variety of techniques. As a result, in situations where computations are feasible for extremal ideals, we provide concrete sharp bounds for many of these invariants. Our methods reduce finding homological invariants and algebraic constructions for infinitely many ideals to computations for a single highly symmetric ideal, based solely on the number of generators.