BCAM Working Seminar APDE: Connections between Bombieri-type inequalities and equidistribution of points on Riemannian manifolds

Abstract
LetP1, . . . ,Pm∈K[x1, . . . ,xn]be a set of polynomials withK∈{C,R}and let||be a norm defined inK[x1, . . . ,xn]. The following problem arises naturallyin the context of the measures of polynomials: define a constantAdependingonly on the degrees of the polynomialsP1, . . . ,Pmand such that(1)||P1||. . .||Pm||≤A||P1. . .Pm||.This inequality has been proposed for many different norms and the literatureversing on this problem is rather extensive. Authors in[1]consider the Bombierinorm (sometimes also called Kostlan norm or Weyl norm) and provide with aconstantAfor the case of polynomials with complex coefficients that turns out tobe sharp form=2. In the recent paper[2]we explore the connections betweenequation (1) for univariate polynomials with complex coefficients and a set ofevenly distributed spherical points, more concretely, minimizers of the discretelogarithmic energy on the sphereS2. On a joint work with Hökan Hedenmalmand Joaquim Ortega-Cerdá[3]we rephrase the inequality proposed in[2]onan integral form on the sphere and propose a new family of inequalities inspiredon equation (1) that can be stated for any compact Riemannian manifold.Throughout this talk, we will present the different results found in articles[1-3], briefly commenting on some of their proofs and making special emphasison the geometric intuition behind them.REFERENCES[1]B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery,Products of polynomials in manyvariables, Journal of Number Theory36(1990), no. 2, 219 -245.↑1[2]U. Etayo,A sharp bombieri inequality, logarithmic energy and well conditioned polynomials,ArXiv: 1912.05521 (2019).↑1[3]H. Hedenmalm U. Etayo and J. Ortega-Cerdá,Work in progress(2020).↑1