Indecomposability for differential algebraic groups

TL;DR

This paper introduces a new notion of indecomposability in differential algebraic groups, proves an indecomposability theorem, and applies it to establish structural and definability results, including the perfection of certain groups and bounds on Kolchin polynomials.

Contribution

It develops a novel indecomposability criterion for differential algebraic groups and applies it to solve open problems and derive new bounds in differential algebraic geometry.

Findings

Noncommutative almost simple differential algebraic groups are perfect.

Established bounds on Kolchin polynomials of generators.

Proved an indecomposability theorem for differential algebraic groups.

Abstract

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of subvarieties to be a differential algebraic subgroup. This theorem is used for various definability results. For instance, we show every noncommutative almost simple differential algebraic group is perfect, solving a problem of Cassidy and Singer. We also establish numerous bounds on Kolchin polynomials, some of which seem to be of a nature not previously considered in differential algebraic geometry; in particular, we establish bounds on the Kolchin polynomial of the generators of the differential field of definition of a differential algebraic variety.