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arXiv · 2507.07976

Pierce-Birkhoff conjecture is true for splines

Zehua Lai·Lek-Heng Lim
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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.

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Zehua Lai, Lek-Heng Lim. 2025-07-10. Pierce-Birkhoff conjecture is true for splines. https://arxiv.org/abs/2507.07976

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