# 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…

## References

SHOWING 1-8 OF 8 REFERENCES
Lattice-ordered rings and function rings.
• Mathematics
• 1962
Introduction: This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings—the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two
Expansions of the Real Field with Power Functions
• Chris Miller
• Mathematics, Computer Science
Ann. Pure Appl. Log.
• 1994
It is shown that the (O-minimal) theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization.
The Basic Theory of Power Series
I Power Series.- 1 Series of Real and Complex Numbers.- 2 Power Series.- 3 Ruckert's and Weierstrass's Theorems.- II Analytic Rings and Formal Rings.- 1 Mather's Preparation Theorem.- 2 Noether's
Some comments and examples on generation of (hyper-)archimedean $\ell$-groups and $f$-rings
• Mathematics
• 2010
Dans cet article, nous donnons les bases d'une theorie sur la generation des l-groupes et des f-anneaux reduits a partir de certaines sous-structures. Nous sommes concernes en premier lieu par les
Birkhoff (G.) and Pierce (R.S.)
• 1956
vol
• 248, Cambridge Univ. Press,
• 1998
Lattice-ordered Rings
• Mathematics
• 1940