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Workshop

On the Number of Real Zeros of Random Sparse Polynomial Systems

  • Josué Tonelli-Cueto (Johns Hopkins University, USA)
Felix-Klein-Hörsaal Universität Leipzig (Leipzig)

Abstract

Consider a polynomial system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ real polynomials in $n$ variables, where each $f_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Kushnirenko's conjecture asks whether the number of positive zeros of such a polynomial system is of the form $\mathrm{poly}(t_1,\ldots,t_n)$. As of today, this is one of the most challenging problems in real algebraic geometry, being open even in the two-dimensional case!
In this talk, we will present a probabilistic version of Kushnirenko's conjecture. We will show, among other variations of the result, that if the coefficients are independent Gaussian of any variance, then the expected number of positive zeros of the random system is bounded from above by $4^{-n} \prod_{k=1}^n t_k(t_k-1)$.

conference
29.07.24 02.08.24

MEGA 2024

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Christian Lehn

Ruhr-Universität Bochum

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Rainer Sinn

Universität Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Simon Telen

Max Planck Institute for Mathematics in the Sciences

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