### Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.

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Let $h:{\mathbb{R}}^{n}\to \mathbb{R}$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_{i}{inf}_{j}{f}_{ij}$, for some finite collection of polynomials ${f}_{ij}\in \mathbb{R}[{x}_{1},...,{x}_{n}]$. (A simple example is $h\left({x}_{1}\right)=\left|{x}_{1}\right|=sup\{{x}_{1},-{x}_{1}\}$.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers;...

In this note it is presented a new rational and continuous solution for Hilbert's 17th problem, which asks if an everywhere positive polynomial can be expressed as a sum of squares of rational functions. This solution (Theorem 1) improves the results in [2] in the sense that our parametrized solution is continuous and depends in a rational way on the coefficients of the problem (what is not the case in the solution presented in [2]). Moreover our method simplifies the proof and it is easy to generalize...

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