Quantitative equidistribution for the solutions of a system of sparse polynomial equations

Details Quantitative equidistribution for the solutions of a system of sparse polynomial equations Quantitative equidistribution for the solutions of a system of sparse polynomial equations

2012-03-08

arxiv.org/abs/1203.1843v1

Mathematics

Complex Variables

29 pages, 2 figures

Scientific paper

For a sequence of n Laurent polynomials in n variables with complex coefficients such that the size of the coefficients is not too big with respect to the facet resultants of the input sequence, we show that the solutions in the algebraic torus of the system of equations defined by this sequence, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case, up to a exponent, 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 the integers, and of random systems of Laurent polynomials over the field of complex numbers.

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