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Journal Articles American Journal of Mathematics Year : 2014

Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

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.
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 coefficients
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Dates and versions

hal-00918250 , version 1 (13-12-2013)

Identifiers

  • HAL Id : hal-00918250 , version 1

Cite

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