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.