Density of Complex Critical Points of a Real Random SO(m+1) Polynomials

Abstract

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper [Mac09], the author used the Poincare- Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac- Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.

15 pages

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• 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
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