Density of Complex Zeros of a System of Real Random Polynomials

Details Density of Complex Zeros of a System of Real Random Polynomials Density of Complex Zeros of a System of Real Random Polynomials

2010-06-20

arxiv.org/abs/1006.3932v1

Macdonald, Brian. Density of Complex Zeros of a System of Real Random Polynomials. Journal of Statistical Physics: Volume 136,

Physics

Mathematical Physics

25 pages, 4 figures

Scientific paper

10.1007/s10955-009-9810-5

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 $m\geq 2$ case than in the $m=1$ case: the density goes to infinity instead of tending linearly to zero.

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