On the constancy regions for mixed test ideals

TL;DR

This paper investigates how mixed test ideals partition the non-negative real space, showing each region corresponds to a preimage of a p-fractal function and illustrating their complex shapes beyond rational polytopes.

Contribution

It introduces a novel description of constancy regions for mixed test ideals using p-fractal functions, revealing their potentially intricate geometric structure.

Findings

Regions are preimages of natural numbers under p-fractal functions

Regions may not be finite unions of rational polytopes

Provides examples illustrating complex geometric structures

Abstract

In this note we study the partition of $\mathbb{R}_{\geq0}^{n}$ given by the regions where the mixed test ideals $\tau(\mathfrak{a}_{1}^{t_{1}}... \mathfrak{a}_{n}^{t_{n}})$ are constant. We show that each region can be described as the preimage of a natural number under a p-fractal function $\varphi:\mathbb{R}_{\geq0}^{n}\rightarrow\mathbb{N}$. In addition, we give some examples illustrating that these regions do not need to be composed of finitely many rational polytopes.