Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem
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A01=Daniel Perrucci
A01=Henri Lombardi
A01=Marie-Francoise Roy
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Author_Daniel Perrucci
Author_Henri Lombardi
Author_Marie-Francoise Roy
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Product details
- ISBN 9781470441081
- Weight: 252g
- Dimensions: 178 x 254mm
- Publication Date: 30 Apr 2020
- Publisher: American Mathematical Society
- Publication City/Country: US
- Product Form: Paperback
- Language: English
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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^{k}}} } } $ 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^{k}}}+ s^{2^{k}}\max\{2, d\}^{16^{k}{\mathrm bit}(d)} \right)} } $ 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.
Henri Lombardi, Universite de Franche-Comte, Besancon, France
Daniel Perrucci, Universidad de Buenos Aires, Argentina
Marie-Francoise Roy, Universite de Rennes, France
Daniel Perrucci, Universidad de Buenos Aires, Argentina
Marie-Francoise Roy, Universite de Rennes, France
