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On the algebraic structure of differentially homogeneous polynomials

Antoine Etesse

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).

On the algebraic structure of differentially homogeneous polynomials

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).

Paper Structure

This paper contains 23 sections, 20 theorems, 141 equations.

Table of Contents

  1. Differentially homogeneous polynomials
  2. Definitions and first properties
  3. Definitions
  4. Factoriality
  5. Reformulation via the formalism of Invariant Theory
  6. An alternative proof of the Schmidt--Kolchin conjecture
  7. A simple, but important, Lemma
  8. The alternative proof
  9. Link with harmonic polynomials
  10. Invariant tensors and harmonic polynomials
  11. DeConcini--Procesi ideals, and $\mu$-harmonic polynomials
  12. Notations and definitions
  13. The ideal $I_{k}$ is a DeConcini--Procesi ideal
  14. A structural result concerning $\mu$-harmonic polynomials
  15. General strategy to study $\mu$-harmonic polynomials
  16. ...and 8 more sections

Key Result

Theorem 1

The graded algebra $(V^{\mathop{\mathrm{diff}}\nolimits})^{(k)}$ is finitely generated by $(N+1)$ generators in degree $1$ and order $0$ and, for $2 \leq i \leq k+1$, Furthermore, the set of generators is explicit, and minimal.

Theorems & Definitions (49)

  • Theorem 1: Main Theorem
  • Corollary 2
  • Theorem 3: Reformulation of the Main Theorem
  • Definition 1.1.1: Differential polynomials
  • Definition 1.1.2: Differentially homogeneous polynomials
  • Lemma 1.1.3
  • proof
  • Corollary 1.1.4
  • proof
  • Remark 1.2.1
  • ...and 39 more