Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces ${\mathbb{X}}^{n-1}\subset {\mathbb{P}}^n(\mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the ‘celestial’ horizon lies near $d⩾2n$.

For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain $$d⩾(\sqrt{n}\log n{)}^n,$$ and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain $$d⩾{(n\log n)}^n.$$ The latter improves $d⩾n^{2n}$ obtained by Merker in arxiv.org/1807/11309/.

Admitting a certain technical conjecture $I_0⩾{\tilde{I}}_0$, the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau) conducts to constant power $n$, namely to $$d⩾2^{5n}\text{and, respectively, to}d⩾4^{5n}.$$

In Spring 2021, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0⩾{\tilde{I}}_0$ a conjecture which will be established up to dimension $n=50$.