On the Schmidt--Kolchin conjecture on differentially homogeneous polynomials. Applications to (twisted) jet differentials on projective spaces

TL;DR

This paper proves the Schmidt--Kolchin conjecture regarding the dimension of differentially homogeneous polynomials and explores their connection to twisted jet differentials on projective spaces, providing explicit descriptions.

Contribution

It establishes the conjecture and links differentially homogeneous polynomials to twisted jet differentials, offering new insights into their structure on projective spaces.

Findings

Proof of the Schmidt--Kolchin conjecture confirming the dimension formula

Establishment of a correspondence between differentially homogeneous polynomials and twisted jet differentials

Explicit understanding of twisted jet differentials on projective spaces

Abstract

The main goal of this paper is to prove the Schmidt--Kolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a one-to-one correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a by-product of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.