Polynomials and entire functions with real zeros

Author(s)

Fábio Rodrigues Lucas

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2010-06-15

Examining board members

Dimitar Kolev Dimitrov; Eliana Xavier Linhares de Andrade; Sergio Antonio Tozoni; Claudio Aguinaldo Buzzi; Alagacone Sri Ranga

• Advisor: Dimitar Kolev Dimitrov

Abstract

In this thesis we approach problems concerning zeros of polynomials and entire functions. We establish explicit formula for the polynomial in the Sturm sequence, generated by a polynomial and its derivative. As a consequence, we obtain necessary and sufficient conditions for a polynomial without multiple zeros to possess only real zeros. We prove also the truth of certain necessary conditions for the Riemann Hypothesis, thus extending a previous result of Csordas, Norfolk and Varga who established a conjecture of Pôlya (AU)