An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem

The authors prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely they express a nonnegative polynomial as a sum of squares of rational functions and obtain as degree estimates for the numerators and denominators the following tower of five exponentials 2...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Lombardi, Henri
Otros Autores: Perrucci, Daniel, Roy, Marie-Françoise
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Providence : American Mathematical Society, 1920.
Colección:Memoirs of the American Mathematical Society Ser.
Temas:
Polynomials.
Algebraic fields.
Recursive functions.
Algebraic stacks.
Polynômes.
Corps algébriques.
Fonctions récursives.
Algebraic stacks
Algebraic fields
Polynomials
Recursive functions
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title page
  • Chapter 1. Introduction
  • 1.1. Hilbert's 17th problem
  • 1.2. Positivstellensatz
  • 1.3. Historical background on constructive proofs and degree bounds
  • 1.4. Our results
  • 1.5. Organization of the paper
  • Acknowledgements
  • Chapter 2. Weak inference and weak existence
  • 2.1. Weak inference
  • 2.2. Weak existence
  • 2.3. Complex numbers
  • 2.4. Identical polynomials
  • 2.5. Matrices
  • Chapter 3. Intermediate value theorem
  • 3.1. Intermediate value theorem
  • 3.2. Real root of a polynomial of odd degree
  • Chapter 4. Fundamental theorem of algebra
  • 4.1. Fundamental theorem of algebra
  • 4.2. Decomposition of a polynomial into irreducible real factors
  • 4.3. Decomposition of a polynomial into irreducible real factors with multiplicities
  • Chapter 5. Hermite's Theory
  • 5.1. Signature of Hermite's quadratic form and real root counting
  • 5.2. Signature of Hermite's quadratic form and signs of principal minors
  • 5.3. Sylvester Inertia Law
  • 5.4. Hermite's quadratic form and Sylvester Inertia Law
  • Chapter 6. Elimination of one variable
  • 6.1. Thom encoding of real algebraic numbers
  • 6.2. Conditions on the parameters fixing the Thom encoding
  • 6.3. Conditions on the parameters fixing the real root order on a family
  • 6.4. Realizable sign conditions on a family of polynomials
  • Chapter 7. Proof of the main theorems
  • Chapter 8. Annex
  • Bibliography
  • Back Cover

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