Computing real powers of monomial ideals

TL;DR

This paper introduces algorithms for computing real powers of monomial ideals, extending traditional integer powers to positive real exponents, and reveals that these powers form a step function with rational jump points.

Contribution

It generalizes the concept of ideal powers to real exponents, connecting to convex polytopes and providing algorithms for their computation.

Findings

Real powers of monomial ideals form a step function with rational jumps.

The problem of computing real powers reduces to rational exponents.

The work links ideal powers to convex polytope theory.

Abstract

This paper concerns the exponentiation of monomial ideals. While it is customary for the exponentiation operation on ideals to consider natural powers, we extend this notion to powers where the exponent is a positive real number. Real powers of a monomial ideal generalize the integral closure operation and highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the real powers of a monomial ideal. An important result is that given any monomial ideal $I$, the function taking real numbers to the corresponding real power of $I$ is a step function whose jumping points are rational. This reduces the problem of determining real powers to rational exponents.