Polinômios homogêneos não analíticos e uma aplicação às séries de Dirichlet

Abstract

We study continuous homogeneous polynomials that are not analytic. The main results in this work refer to the existence of linear structures formed by non-analytical polynomials, as well as an application in which these polynomials are used in order to solve Bohr’s Absolute Abscissa Problem. To do this we begin with the study of homogeneous polynomials between Banach spaces and their main properties. Then we see in detail the construction of the 2-homogeneous polynomial constructed by Toeplitz and the m-homogeneous polynomial of Bohnenblust and Hille. With the help of these polynomials we will construct a linear subspace isomorphic to the Banach space ℓ1 , formed by homogeneous polynomials that are not analytic on a given vector, in particular we will have that the set of homogeneous non-analytic polynomials in c0 is spaceable. As an application we will see the Bohr Absolute Convergence Problem, which consists in determining the maximum distance between the abscissa of absolute and uniform convergence of a Dirichlet series, having as a useful tool for solution the Bohnenblust and Hille polynomial.