Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

For a system of Laurent polynomials $f_1,\ldots,f_n\in{\Bbb C}[x_1^{\pm1},\ldots,x_n^{\pm1}]$ whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic torus $({\Bbb C}^{\times})^n$ of the system of equations $f_1=\cdots=f_n=0$, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erd\"os 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 ${\Bbb Z}$ and of random systems of Laurent polynomials over ${\Bbb C}$.