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Chapter 1 Representation of Positive Polynomials

  • Roberto Cominetti
  • Francisco Facchinei
  • Jean B. Lasserre
Chapter
Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Abstract

In one dimension, the ring \({\mathbb{R}} [x]\) of real polynomials in a single variable has the fundamental property (Theorem 1.6) that every nonnegative polynomial \(p \in \mathbb{R} [x]\) is a sum of squares of polynomials, that is, \(p(x) \geq 0, \forall_{x} \in \mathbb{R}\,\,\,\Longleftrightarrow\,\,\,p(x) = \sum\limits_{i=1}^{k} h_i(x)^{2}, \forall_{x} \in \mathbb{R}\), for finitely many polynomials (h i).

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Roberto Cominetti
    • 1
  • Francisco Facchinei
    • 2
  • Jean B. Lasserre
    • 3
  1. 1.Departamento de Ingeniería IndustrialUniversidad de ChileSantiago de ChileChile
  2. 2.Department of Computer, Control, and Management Engineering Antonio RubertiUniversità di Roma “La Sapienza”RomaItaly
  3. 3.LAAS-CNRSToulouseFrance

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