An elementary recursive bound for effective positivstellensatz and hilbert's 17th problem
著者
書誌事項
An elementary recursive bound for effective positivstellensatz and hilbert's 17th problem
（Memoirs of the American Mathematical Society, no. 1277）
American Mathematical Society, c2020
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注記
"January 2020, volume 263, number 1277 (seventh of 7 numbers)"
Includes bibliographical reference (p. 123-125)
内容説明・目次
内容説明
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.
目次
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.
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