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

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Title: Density of Complex Critical Points of a Real Random SO( m+1) Polynomial.
Authors: Macdonald, Brian1,2 bmac@jhu.edu
Source: Journal of Statistical Physics. Nov2010, Vol. 141 Issue 3, p517-531. 15p. 2 Graphs.
Subjects: Random polynomials, Complex variables, Critical point (Thermodynamics), Scaling laws (Statistical physics), Phase equilibrium, Density functionals
Abstract: We study the density of complex critical points of a real random SO( m+1) polynomial in m variables. In a previous paper (Macdonald in J. Stat. Phys. 136(5):807, ), the author used the Poincaré-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. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Density of Complex Critical Points of a Real Random SO( m+1) Polynomial.
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  Data: <searchLink fieldCode="DE" term="%22Random+polynomials%22">Random polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Complex+variables%22">Complex variables</searchLink><br /><searchLink fieldCode="DE" term="%22Critical+point+%28Thermodynamics%29%22">Critical point (Thermodynamics)</searchLink><br /><searchLink fieldCode="DE" term="%22Scaling+laws+%28Statistical+physics%29%22">Scaling laws (Statistical physics)</searchLink><br /><searchLink fieldCode="DE" term="%22Phase+equilibrium%22">Phase equilibrium</searchLink><br /><searchLink fieldCode="DE" term="%22Density+functionals%22">Density functionals</searchLink>
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  Data: We study the density of complex critical points of a real random SO( m+1) polynomial in m variables. In a previous paper (Macdonald in J. Stat. Phys. 136(5):807, ), the author used the Poincaré-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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10955-010-0057-y
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      – SubjectFull: Complex variables
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      – SubjectFull: Critical point (Thermodynamics)
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      – SubjectFull: Phase equilibrium
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