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Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Charles N. Delzell (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Let h : n 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 i j , for some finite collection of polynomials f i j [ x 1 , ... , x n ] . (A simple example is h ( x 1 ) = | x 1 | = sup { x 1 , - x 1 } .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for n 2 ; it remains open for n 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;...

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