Extension of the twovariable PierceBirkhoff conjecture to generalized polynomials
Abstract
Let R denote the reals, and let h: R^n > R be a continuous, piecewisepolynomial function. The PierceBirkhoff 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 eprints
 Pub Date:
 January 2010
 arXiv:
 arXiv:1002.0038
 Bibcode:
 2010arXiv1002.0038D
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Logic;
 14P15 (Primary);
 03C64;
 06B25;
 26B99;
 26C05 (Secondary)
 EPrint:
 16 pages, 4 figures