Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

André Galligo 1 Carlos d'Andrea 2 Martin Sombra 3
1 GALAAD2 - Géométrie , Algèbre, Algorithmes
CRISAM - Inria Sophia Antipolis - Méditerranée
3 ICREA \& Universitat de Barcelona
ICREA - Institució Catalana de Recerca i Estudis Avançats
Abstract : For 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.
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Journal articles
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https://hal.inria.fr/hal-00918250
Contributor : André Galligo <>
Submitted on : Friday, December 13, 2013 - 10:44:40 AM
Last modification on : Friday, August 2, 2019 - 3:06:02 PM

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André Galligo, Carlos d'Andrea, Martin Sombra. Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations. American Journal of Mathematics, Johns Hopkins University Press, 2014. ⟨hal-00918250⟩

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