On the number of real roots of random polynomials

On the Number of Real Roots of Random Polynomials

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős-Offord, showed that the expectation of the number of real roots is 2 π logn + o(logn). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 π...

On the Number of Real Roots of Polynomials

Our main theorem, proved in § 2 establishes some of the properties of F(x, y) = 0 when we drop all restrictions on / and require only that all the roots of h be real (of arbitrary sign). In the last section, we apply this theorem to obtain an extension of Theorem 1.1 which states that, for h restricted as in Pόlya's theorem, there are at least as many intersection points as the number of real r...

On the average number of sharp crossings of certain Gaussian random polynomials

On Roots of Random Polynomials

We study the distribution of the complex roots of random polynomials of degree n with i.i.d. coefficients. Using techniques related to Rice’s treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we ...

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...