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.
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