Pierce-Birkhoff conjecture is true for splines

2025/07/10 by Lai, Zehua, Lim, Lek-Heng
#13J25 #14P10 #26B40 #41A15 #52B55 #52C35 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.07976

Abstract

We prove the Pierce--Birkhoff conjecture for splines, i.e., continuous piecewise polynomials of degree d in n variables on a hyperplane partition of ℝⁿ, can be written as a finite lattice combination of polynomials. We will provide a purely existential proof, followed by a more in-depth analysis that yields effective bounds.

Citations

• Attention is a smoothed cubic spline
• Bivariate Semialgebraic Splines
• Semialgebraic Splines
• On the Pierce-Birkhoff Conjecture
• Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
• Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials
• On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
• Max-Min Representation of Piecewise Linear Functions

Related

• Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials
• On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
• Multivariate expansivity theory and Pierce-Birkhoff conjecture
• On the strong separation conjecture
• On the Pierce-Birkhoff Conjecture