Pierce-Birkhoff conjecture is true for splines

TL;DR

This paper proves the Pierce-Birkhoff conjecture for splines, showing that continuous piecewise polynomials can be expressed as finite lattice combinations of polynomials, with both existential and effective bounds provided.

Contribution

It establishes the conjecture for splines, offering both an existential proof and effective bounds, advancing understanding of polynomial representations.

Findings

Proof of Pierce-Birkhoff conjecture for splines

Existence of finite lattice combination representations

Effective bounds for polynomial decompositions

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 $\mathbb{R}^n$, 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.