Abstract.We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ... , t") projected onto the surface of the unit sphere, divided by n . The probability density of the real zeros is proportional to how fast this curve is traced out.We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.
Consider a random matrix whose elements are independent random variables from a standard (mean zero, variance one) normal distribution. Unless otherwise stated, we omit the distribution and simply use the term "random matrix" to denote a matrix with independent standard normally distributed elements. Other distributions are considered in §8.Here is one of our main resultsAsymptotic Number of Real Eigenvalues. If En denotes the expected number of real eigenvalues of an n-by-n random matrix, thenAsymptotic Series. As n -+ 00 ,Let A be a 50 x 50 random matrix. Figure 1 on the following page plots normalized eigenvalues AIV50 in the complex plane for fifty matrices. Thus there are 2500 dots in the figure. There are a number of striking features in the diagram. First, nearly all the normalized eigenvalues appear to fall in the interior of the unit disk. This is Girko's as yet unverified circular law [12], which states that as n gets large, AI vn is uniformly distributed in the unit disk. It follows that the proportion of eigenvalues on the real line (also strikingly visible to the eye) must tend to 0 as n -+ 00. Our results show how fast this converges. A simpler version of this circular law occurs when the random matrix has elements from a complex normal distribution; i.e. the real and imaginary parts of each element are independent standard normals. In this case the exact distribution for the eigenvalue distribution and radius can be found in Ginibre [11] and is reported by Mehta [24, p. 300] and Hwang [17]. In this case, the squares of the absolute values of the eigenvalues are independent random
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. Contents Part I-Introduction 1. Overview 2. Summary of previous results 2.1. Polynomials with independent standard normal coefficients 2.2. The most natural random polynomial 2.3. Random harmonic polynomials 2.3. Rojas polynomials 3. Level of generality 3.1. Homogeneous and inhomogeneous systems 3.2. Underdetermined and overdetermined systems 3.3. Mixed systems Part II-Orthogonally invariant normal coefficients 4. Classification of invariant inner products 4.1. The complex analogue 4.2. Classification of indefinite inner products 4.3. Gegenbauer polynomials 4.4. The eigenspaces of r 2 ∇ 2 4.5. Classification of definite inner products 5. Invariant normal random polynomials 5.1. The complex analog 5.2. The expected volume of a random real variety 5.3. Non-central invariant coefficients 5.4. Central invariant coefficients 5.5. Classification of invariant normal measures 6. Quadratic forms and random symmetric matrices
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