On the expected number of real roots of a system of random polynomial equations

We unify and generalize several known results about systems of random polynomials. We first classify all orthogonally invariant normal measures for spaces of polynomial mappings. For each such measure we calculate the expected number of real zeros. The results for invariant measures extend to underdetermined systems, giving the expected volume for orthogonally invariant random real projective varieties. We then consider noninvariant measures, and show how the real zeros of random polynomials behave under direct sum, tensor product and composition.

[1]  Alan Edelman,et al.  How many zeros of a random polynomial are real , 1995 .

[2]  Leboeuf,et al.  Distribution of roots of random polynomials. , 1992, Physical review letters.

[3]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[4]  J. M. Rojas,et al.  On the Average Number of Real Roots of Certain Random Sparse Polynomial Systems , 1996 .

[5]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[6]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[7]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[8]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[9]  S. Smale,et al.  Complexity of Bezout’s Theorem II Volumes and Probabilities , 1993 .

[10]  A. T. Bharucha-Reid,et al.  The Number and Expected Number of Real Zeros of Other Random Polynomials , 1986 .

[11]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[12]  Andrew McLennan,et al.  The expected number of real roots of a multihomogeneous system of polynomial equations , 1999, math/9904120.

[13]  R. Leighton,et al.  The Feynman Lectures on Physics; Vol. I , 1965 .

[14]  On the Roots of Algebraic Equations , 1951 .

[15]  A. T. Bharucha-Reid,et al.  Random Matrices and Random Algebraic Polynomials , 1986 .

[16]  J. M. Hammersley,et al.  The Zeros of a Random Polynomial , 1956 .

[17]  K. Farahmand.  On the Average Number of Real Roots of a Random Algebraic Equation , 1986 .

[18]  A. Edelman,et al.  How many eigenvalues of a random matrix are real , 1994 .

[19]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[20]  J. Maurice Rojas,et al.  Random Sparse Polynomial Systems , 2000 .