Global sections of the positively twisted Green-Griffiths bundles

With various jet orders $k$ and weights $n$, let $E_{k,n}^{\rm GG}$ be the Green-Griffiths bundles over the projective space $\mathbb{P}^N (\mathbb{C})$. Denote by $\mathcal{O} (d)$ the tautological line bundle over $\mathbb{P}^N (\mathbb{C})$. Although only negative twists are of interest for applications to complex hyperbolicity (above general type projective submanifolds $Y \subset \mathbb{P}^N (\mathbb{C})$), it is known that the positive twists $E_{k,n}^{\rm GG} \otimes \mathcal{O} (d)$ enjoy nontrivial global sections. In this article, we establish that for every $d \geqslant 1$ and for every jet order $k \geqslant d-1$: $$\dim\, H^0 \bigg( \mathbb{P}^N,\,\, \bigoplus_{n=1}^{\infty} E_{k, n}^{\text{GG}} \otimes \mathcal{O}(d) \bigg) = (N+1)^d.$$ This theorem is actually a corollary of a recent work of Etesse, devoted to a proof, from the point of view of differentially homogeneous polynomials, of the so-called Schmidt-Kolchin-Reinhart conjecture, by means of (advanced) Representation Theory. As Etesse discovered a (simple) tight link with the Green-Griffiths bundles, both statements are in fact equivalent. Our objective is to set up an alternative proof of the above precise dimension estimate, from the Green-Griffiths point of view (only). More precisely, we find an explicit description of all concerned global sections. Our arguments are elementary, and use only determinants, linear algebra, monomial orderings. One old hope is to discover some explicit formulas for global sections of negatively twisted Green-Griffiths bundles over projective general type submanifolds $Y \subset \mathbb{P}^N (\mathbb{C})$, a problem still open.