Multivariate expansivity theory and Pierce-Birkhoff conjecture

TL;DR

This paper extends single-variable expansivity theory to multivariate cases and applies it to develop inequalities relevant to the Pierce-Birkhoff conjecture, advancing understanding in real algebraic geometry.

Contribution

It introduces a multivariate expansivity framework and applies it to inequalities related to the Pierce-Birkhoff conjecture, a novel extension of existing theory.

Findings

Developed a multivariate expansivity theory for polynomial tuples.

Formulated inequalities to analyze the Pierce-Birkhoff conjecture.

Provided new tools for studying real algebraic geometry problems.

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.