  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
$d \geqslant 2n$. For Green-Griffiths algebraic degeneracy of entire
holomorphic curves, we obtain: \[ d \,\geqslant\, \big(\sqrt{n}\,{\sf
log}\,n\big)^n, \] and for Kobayashi-hyperbolicity (constancy of entire
curves), we obtain: \[ d \,\geqslant\, \big(n\,{\sf log}\,n\big)^n. \] The
latter improves $d \geqslant n^{2n}$ obtained by Merker in
arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0 \geqslant
\widetilde{I}_0$, the method employed (Diverio-Merker-Rousseau, B\'erczi,
Darondeau) conducts to constant power $n$, namely to: \[ d\ ,\geqslant\, 2^{5n}
\qquad \text{and, respectively, to:} \qquad d \,\geqslant\, 4^{5n}. \] In
Spring 2019, a forthcoming prepublication based on intensive computer
explorations will present several subconjectures supporting the belief that
$I_0 \geqslant \widetilde{I}_0$, a conjecture which will be established up to
dimension $n = 50$.
