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 and efficient sufficient condition for the existence of a solution. 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.

Bio-Sketch: Professor Mark Rudelson received his Ph.D. from Hebrew University of Jerusalem under the supervision of Professor Joram Lindenstrauss. He is currently at the University of Michigan. He is on the editorial boards of Journal of Theoretical Probability and Michigan Mathematical Journal. His research interests are asymptotic geometric analysis, high-dimensional probability, and convex geometry.