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

ISBN-13: 9781470441081

Henri Lombardi;Daniel Perrucci;Marie-Francoise Roy

Opis

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^{ 2^{ 2^{d^{4^}} } } $ where $d$ is the number of variables of the input polynomial. The authors' method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely the authors give an algebraic certificate of the emptyness of the realization of a system of sign conditions and obtain as degree bounds for this certificate a tower of five exponentials, namely $ 2^{ 2^{left(2^{max{2,d}^{4^}}+ s^{2^}max{2, d}^{16^{mathrm bit}(d)} ight)} } $ where $d$ is a bound on the degrees, $s$ is the number of polynomials and $k$ is the number of variables of the input polynomials.

Spis treści

• Introduction
• Weak inference and weak existence
• Intermediate value theorem
• Fundamental theorem of algebra
• Hermite's theory
• Elimination of one variable
• Proof of the main theorems
• Bibliography/References.