Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

International audienceFor a system of Laurent polynomials f1 , . . . , fn ∈ C[x_1^±1 , . . . , x_n^±1 ] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic torus (C× )^n of the system of equations f1 = * * * = fn = 0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdo ̈s and Tur ́an on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials over Z and of random systems of Laurent polynomials over C.On generalise, pour un systeme d'equations polynomiales en plusieurs variables, un resultat classic de Erdos-Turan, sur la localisation asymptotique (quand le degre tend vers l'infini) des racines d'un polynome, en fonction d'une borne sur la taille des coefficient