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

2008/10/27 by Sven Wagner, Wagner, Sven · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0810.4800

openalex publication_date 2008/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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