Multivariate expansivity theory and Pierce-Birkhoff conjecture

Abstract

Motivated by the Pierce-Birkhoff conjecture, we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials in the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$. As an application, we develop some class of inequalities to study the Pierce-Birkhoff conjecture.