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

TL;DR

This paper extends the Pierce-Birkhoff conjecture to generalized polynomials with real exponents in the positive orthant, proving the representation as sup-inf of polynomials for dimensions less than three.

Contribution

It proves an analogous Pierce-Birkhoff type result for generalized polynomials in the positive orthant for n<3, expanding the conjecture's scope beyond traditional polynomials.

Findings

Proved the conjecture for generalized polynomials with real exponents in 2D.

Established the representation as sup-inf of polynomials for these functions.

Focused on the positive orthant where all variables are positive.

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.