On the algebraic structure of differentially homogeneous polynomials

TL;DR

This paper investigates the algebraic structure of differentially homogeneous polynomials, proving it is finitely generated, providing a minimal set of generators, and offering new insights into invariant theory and algebraic geometry.

Contribution

It establishes that the algebra of differentially homogeneous polynomials is finitely generated and presents a minimal generating set, along with a simplified proof of the Schmidt--Kolchin conjecture.

Findings

Proves the algebra is finitely generated.

Provides a minimal set of generators.

Offers new examples in invariant theory.

Abstract

The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way, we provide a simpler proof of the so-called Schmidt--Kolchin conjecture (proved in a previous paper) . From the algebraic point of view, this provides natural compactifications of jet bundles of projective spaces. From the invariant theoretic point of view, this provides new examples, not covered (to our knowledge) by known conjectures in the subject, of unipotent sub-groups of general linear groups, whose algebras of invariants are finitely generated (and more precisely gives a First Fundamental Theorem for such groups, following the terminology in Invariant Theory).