Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials
Abstract
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 < 3; it remains open for n > 2. 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 version, we restrict to the positive orthant, where each x_i > 0. As before, our methods work only for n < 3.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2010
- DOI:
- 10.48550/arXiv.1002.0038
- arXiv:
- arXiv:1002.0038
- Bibcode:
- 2010arXiv1002.0038D
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Logic;
- 14P15 (Primary);
- 03C64;
- 06B25;
- 26B99;
- 26C05 (Secondary)
- E-Print:
- 16 pages, 4 figures