The Pierce-Birkhoff Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A semialgebraic subset of $\mathbb R^n$ is obtained from finitely many polynomial equalities and inequalities by finitely many Boolean operations. A function $f:\mathbb R^n\to\mathbb R$ is piecewise polynomial in this problem if there are closed semialgebraic sets $C_1,\ldots,C_r$ covering $\mathbb R^n$ and polynomials $p_1,\ldots,p_r\in\mathbb R[X_1,\ldots,X_n]$ such that $f|_{C_i}=p_i|_{C_i}$. In particular, $f$ is continuous; arbitrary discontinuous piecewise formulas are excluded.

Statement recorded in the supplied source.

For every $n\ge1$ and every such continuous piecewise-polynomial function $f:\mathbb R^n\to\mathbb R$, do there exist a positive integer $s$, positive integers $t_1,\ldots,t_s$, and polynomials $q_{ij}\in\mathbb R[X_1,\ldots,X_n]$ such that, for every $x\in\mathbb R^n$, \[ f(x)=\max_{1\le i\le s}\ \min_{1\le j\le t_i}q_{ij}(x)? \] [S2]

Short English statement

Can every continuous piecewise-polynomial real function be expressed as a finite maximum of finite minima of polynomials?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1181 (ALG-021). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 François Lucas, James Madden, Daniel Schaub and Mark Spivakovsky, Approximate roots of a valuation and the Pierce--Birkhoff Conjecture, arXiv:1003.1188, Section 0.1, Definition 0.1.1 and Conjecture 0.1.2. https://arxiv.org/abs/1003.1188.