Semi-stable Reduction Implies Minimality of the Resultant

TL;DR

This paper proves that for dynamical systems on projective space over number or function fields, semi-stable reduction guarantees the minimality of the resultant, leading to globally minimal presentations over number fields.

Contribution

It establishes a link between semi-stable reduction and minimality of the resultant, and shows that globally minimal presentations exist over number fields.

Findings

Semi-stable reduction implies minimality of the resultant.

Every dynamical system over a number field admits a globally minimal presentation.

Abstract

For a dynamical system on n-dimensional projective space over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number field admits a globally minimal presentation.