Section: New Results

Symbolic-Numeric Analysis

Cubatures, and related problems, with symmetry

Participants : Mathieu Collowald, Evelyne Hubert.

We address the computation of cubature formulae as a moment problem. Symmetry by finite groups arise naturally for cubatures. We developed the algebraic results to use the symmetry in order to reduce the number of parameters and the size of the matrices involved in the flat extension.

Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations

Participant : André Galligo.

For a system of Laurent polynomials $f_1,...,f_n\in ℂ[x_1^{\pm 1},...,x_n^{\pm 1}]$ whose coefficients are not too big with respect to its directional resultants, we show in [6] that the solutions in the algebraic torus ${\left({ℂ}^{*}\right)}^n$ 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 Erdös and Turán 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 $\mathbb{Z}$ and of random systems of Laurent polynomials over $ℂ$.

Joint work with Carlos D'Andrea (DM-UBA - Departamento de Matemática, Spain), Martin Sombra (ICREA & Universitat de Barcelona, Spain).