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Once first answers in any dimension to the Green-Griffiths and Kobayashiconjectures 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 entireholomorphic curves, we obtain: \[ d \,\geqslant\, \big(\sqrt{n}\,{\sflog}\,n\big)^n, \] and for Kobayashi-hyperbolicity (constancy of entirecurves), we obtain: \[ d \,\geqslant\, \big(n\,{\sf log}\,n\big)^n. \] Thelatter improves $d \geqslant n^{2n}$ obtained by Merker inarxiv.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: \[ d\ ,\geqslant\, 2^{5n}\qquad \text{and, respectively, to:} \qquad d \,\geqslant\, 4^{5n}. \] InSpring 2019, a forthcoming prepublication based on intensive computerexplorations will present several subconjectures supporting the belief that$I_0 \geqslant \widetilde{I}_0$, a conjecture which will be established up todimension $n = 50$.

Degrees $d \geqslant \big( \sqrt{n}\, \log\, n\big)^n$ and $d \geqslant  \big( n\, \log\, n\big)^n$ in the Conjectures of Green-Griffiths and of  Kobayashi