Multiplicity in difference geometry

TL;DR

This paper establishes a fundamental principle for preserving multiplicity in difference geometry, enabling the development of a broader intersection theory and showing uniform fiber sizes in difference curve morphisms.

Contribution

It introduces a principle ensuring multiplicity preservation in difference geometry, facilitating the advancement of intersection theory in this field.

Findings

Fibers of -finite morphisms between difference curves are uniformly sized with multiplicities.

Provides foundational results for intersection theory in difference geometry.

Lays groundwork for future research in difference algebraic geometry.

Abstract

We prove a first principle of preservation of multiplicity in difference geometry, paving the way for the development of a more general intersection theory. In particular, the fibres of a \sigma-finite morphism between difference curves are all of the same size, when counted with correct multiplicities.