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An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem (Memoirs of the American Mathematical Society)

Algebra

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About the Author Henri Lombardi, Universite de Franche-Comte, Besancon, FranceDaniel Perrucci, Universidad de Buenos Aires, ArgentinaMarie-Francoise Roy, Universite de Rennes, France Product Description 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.

Product Specifications

Format: paperback
ASIN: 147044108X
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Algebra
Domain: Amazon UK
Release Date: 30 January 2020
Listed Since: 18 December 2019

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An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem (Memoirs of the American Mathematical Society)

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