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.
| ISBN: | 9781470441081 |
| Publication date: | 30th April 2020 |
| Author: | Henri Lombardi, Daniel Perrucci, MF Roy |
| Publisher: | American Mathematical Society |
| Format: | Paperback |
| Pagination: | 113 pages |
| Series: | Memoirs of the American Mathematical Society |
| Genres: |
Algebra Algebraic geometry |
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.
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem features in the following genres: Algebra, Algebraic geometry
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem is available in Paperback
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem was written by Henri Lombardi, Daniel Perrucci, MF Roy and published by American Mathematical Society
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem has 113 pages
Yes it is part of Memoirs of the American Mathematical Society series