Mark Rudelson : When a system of real quadratic equations has a solution

Posted by Osama Khalil , part of the Departmental Colloquium.

At

Feb. 16, 2024, 3 p.m.

In

636 SEO

Abstract

The existence and the number of solutions of a system of polynomial equations in n variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple sufficient condition for the existence of a solution which can be efficiently checked. While the problem and the condition are of algebraic nature, the approach lies entirely within the analysis/probability realm and relies on tools from Fourier analysis and concentration of measure. Joint work with Alexander Barvinok.

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