Quantitative equidistribution for the solutions of systems of sparse polynomial equations

2012/03/08 by Carlos D'Andrea, D'Andrea, Carlos, André Galligo +3
Mathematics · #11K38 #13P15 #30C15 #42B05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Complex Variables (math.CV) #FOS: Mathematics #math.AC #math.AG #math.CV #msc:11K38 #msc:13P15 #msc:30C15 #msc:42B05

paper · pdf · doi:10.48550/arxiv.1203.1843

29 pages, 2 figures. Revised version, accepted for publication in the American Journal of Mathematics

arxiv created 2014/08/06 · arxiv updated 2014/08/07

Abstract

For a system of Laurent polynomials f₁,..., f_n ∈ C[x₁^±1,..., x_n^±1] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f₁=…=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan 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 with integer coefficients, and of random systems of Laurent polynomials with complex coefficients.

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