On the number of real roots of random polynomials

Abstract

Abstract. 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 log n + o(log n). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 π log n + O(1). Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.

Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games

“…Last but not least, as the density functions we analyzed here are closely related to Legendre polynomials, and actually, they are of the same form as the Bernstein polynomials, we have made a clear contribution to the longstanding theory of random polynomials [6,15,45,38](see again discussion in introduction). We have derived asymptotic behaviors for the expected real zeros E B of a random Bernstein polynomial, which, to our knowledge, had not been provided before.…”

“…Theorem 5 and Proposition 2, which are deduced from Theorems 3 and 4, are qualitative statements, which tell us how the expected number of internal equilibria per unit length f 2,d in a d-player twostrategy game changes when the number of players d increases. Furthermore, it is important to note that the expected number of real zeros of a random polynomial has been extensively studied, dating back to 1932 with Block and Pólya's seminal paper [6] (see, for instance, [15] for a nice exposition and [45,38] for the most recent progress). Therefore, our results, in Theorems 2, 3 and 4, provide important, novel insights within the theory of random polynomials, but also reveal its intriguing connections and applications to EGT.…”

“…The study of the distribution and expected number of real zeros of a random polynomial is an active research field with a long history dating back to 1932 with Block and Pólya [6], see for instance [15] for a nice exposition and [45,38] for the most recent results and discussions. Consider a random polynomial of the type…”

Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

“…Further results on the number of real zeroes, including an asymptotic formula for the variance and a central limit theorem, were obtained in the subsequent works by Ibragimov and Maslova [23,35,34]. For more recent results on the number of real roots, see [10,36,11].…”

Persistence probability of a random polynomial arising from evolutionary game theory

“…Our first motivation is from random polynomial theory in which the study of zeros of a random polynomial has been studied extensively since the seminal paper of Block and Pólya [BP32]. We review here relevant work on the persistence probability and refer the reader to standard monographs [BRS86,Far98] and recent articles [TV15,NNV16,DV17,BZ17] and references therein for information on other aspects of random polynomials such as the expected number of roots, central limit theorem and large deviations. A random polynomial can be generally expressed by…”