Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials

@article{Delzell2010ExtensionOT,
  title={Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials},
  author={Charles N. Delzell},
  journal={arXiv: Algebraic Geometry},
  year={2010}
}
Let R denote the reals, and let h: R^n --> 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 R[x_1,...,x_n]. (A simple example is h(x_1) = |x_1| = sup{x_1, -x_1}.) In 1984, L. Mahe and, independently, G. Efroymson, proved this for n 2. In this paper we prove an analogous result for "generalized polynomials" (also known as signomials), i.e… 

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