Charles EDOU NZE

Strongly Polynomial Bounds in Linear Programming and Smale's 9th Problem

This preprint provides an exhaustive 9-page mathematical monograph on Smale's 9th Problem (Steve Smale, 2000), dedicated to the existence of a strongly polynomial-time algorithm for Linear Programming (LP). While the ellipsoid method (Khachiyan, 1979) and projective interior-poin...

Hodge Conjecture via Derived Categories and the HKR Isomorphism

Abstract This memoir presents a proof of the Hodge conjecture on smooth complex projective algebraic varieties. By transposing the topological problem of rational cycles into the existence of objects in the bounded derived category of coherent sheaves Db(X), we show that the conj...

On the Number of Integer Zeros of Straight-Line Polynomials and Smale's 4th Problem

This preprint provides an exhaustive 10-page mathematical monograph on Smale's 4th Problem (Steve Smale, 2000), dedicated to the number of integer and real zeros of polynomials computed by short straight-line programs (arithmetic circuits). Smale's 4th problem asks whether the nu...

On Height Bounds on Diophantine Curves and Smale's 5th Problem

This preprint provides an exhaustive 9-page mathematical monograph on Smale's 5th Problem (Steve Smale, 2000), dedicated to the effective height bounds on rational solutions of Diophantine algebraic curves of genus g ≥ 2. In 1983, Gerd Faltings proved the celebrated Mordell Conje...

On the Finiteness of Relative Equilibria in Celestial Mechanics and Smale's 6th Problem

This preprint provides an exhaustive 8-page mathematical monograph on Smale's 6th Problem (Steve Smale, 2000), dedicated to the finiteness of relative equilibria (planar central configurations) in the Newtonian N-body problem. Smale's 6th problem asks whether the number of relati...

Birch and Swinnerton-Dyer Conjecture via Selmer Complexes and Iwasawa Theory

This document presents a work on the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves defined over ℚ. By employing the theory of Selmer complexes in the sense of Nekovář over the cyclotomic ℤ p-extension ℚ∞/ℚ, we prove the Main Conjecture of Iwasawa Theory for ellip...