Density of Complex Zeros of a System of Real Random Polynomials

Abstract

We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ${\mathbb{R}}^m$ of the system of real random polynomials is different in the $m\ge 2$ case than in the $m=1$ case: the density goes to infinity instead of tending linearly to zero.

25 pages, 4 figures

References in corpus (6)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Universality and scaling of correlations between zeros on complex manifolds
• Quantum Chaotic Dynamics and Random Polynomials
• Critical points and supersymmetric vacua
• Poincare-Lelong approach to universality and scaling of correlations between zeros
• Exact statistics of complex zeros for Gaussian random polynomials with real coefficients

Cited by in corpus (2)

• Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution
• Density of Complex Critical Points of a Real Random SO(m+1) Polynomials