On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

TL;DR

This paper proves the Pierce-Birkhoff Conjecture for smooth two-dimensional affine real algebraic varieties over real closed fields, showing that piecewise polynomial functions can be expressed as suprema of infima of polynomials.

Contribution

It provides a proof of the Connectedness Conjecture for coordinate rings of smooth affine surfaces, establishing the Pierce-Birkhoff Conjecture in this setting.

Findings

Pierce-Birkhoff Conjecture holds for smooth affine surfaces over real closed fields

Connectedness Conjecture is proven for coordinate rings of such surfaces

Piecewise polynomial functions can be represented as suprema of infima of polynomials

Abstract

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.