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

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.