Fix an integer n ≥ 1 and let A = R[x_1,…,x_n]. A subset S ⊆ R^n is semialgebraic if it is a finite union of sets of the form x ∈ R^n : p_1(x) = … = p_k(x) = 0, q_1(x) > 0, …, q_l(x) > 0 with p_1,…,p_k,q_1,…,q_l ∈ A. Call a function h : R^n → R piecewise polynomial if there exist finitely many closed semialgebraic sets P_1,…,P_m ⊆ R^n with R^n = bigcup_i=1^m P_i and polynomials f_1,…,f_m ∈ A such that h(x) = f_i(x) for every i and every x ∈ P_i. Then for every piecewise polynomial function h : R^n → R there exist finite index sets I and J_i (i ∈ I) and polynomials g_ij ∈ R[x_1,…,x_n] for i ∈ I, j ∈ J_i, such that h(x) = max_i ∈ I min_j ∈ J_i g_ij(x) qquad for all x ∈ R^n; that is, the ring of piecewise polynomial functions on R^n coincides with the sublattice of R^R^n generated by A under the operations max and min.

Pierce–Birkhoff conjecture

About 70 years old · traced to 1956

Fix an integer $n \ge 1$ and let $A = \mathbb{R}[x_1,\dots,x_n]$ . A subset $S \subseteq \mathbb{R}^n$ is semialgebraic if it is a finite union of sets of the form

\{x \in \mathbb{R}^n : p_1(x) = \cdots = p_k(x) = 0,\ q_1(x) > 0, \dots, q_l(x) > 0\}

with $p_1,\dots,p_k,q_1,\dots,q_l \in A$ .

Call a function $h : \mathbb{R}^n \to \mathbb{R}$ piecewise polynomial if there exist finitely many closed semialgebraic sets $P_1,\dots,P_m \subseteq \mathbb{R}^n$ with $\mathbb{R}^n = \bigcup_{i=1}^m P_i$ and polynomials $f_1,\dots,f_m \in A$ such that $h(x) = f_i(x)$ for every $i$ and every $x \in P_i$ .

Then for every piecewise polynomial function $h : \mathbb{R}^n \to \mathbb{R}$ there exist finite index sets $I$ and $J_i$ ( $i \in I$ ) and polynomials $g_{ij} \in \mathbb{R}[x_1,\dots,x_n]$ for $i \in I$ , $j \in J_i$ , such that

h(x) = \max_{i \in I} \ \min_{j \in J_i} \ g_{ij}(x) \qquad \text{for all } x \in \mathbb{R}^n;

that is, the ring of piecewise polynomial functions on $\mathbb{R}^n$ coincides with the sublattice of $\mathbb{R}^{\mathbb{R}^n}$ generated by $A$ under the operations $\max$ and $\min$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Pierce–Birkhoff conjecture for inf-sup definable functions
Let $n\in\mathbb{N}$ . A function in $PWP(\mathbb{R}^n)$ is a piecewise polynomial function on $\mathbb{R}^n$ , while $ISD(\mathbb{R}^n)$ denotes the class of inf-sup definable functions introduced in the paper. Both are $f$ -rings, and $ISD(\mathbb{R}^n)\subseteq PWP(\mathbb{R}^n)$ . Pierce–Birkhoff conjecture.
$ISD(\mathbb{R}^n)=PWP(\mathbb{R}^n).$
The conjecture asserts that every piecewise polynomial function on $\mathbb{R}^n$ is inf-sup definable. The supplied text identifies this as the famous Pierce–Birkhoff conjecture but gives no resolution status.
source: S. David Mis, Matti Lassas and Maarten V. de Hoop, “Semialgebraic Neural Networks: From roots to representations”, arXiv:2501.01564 (2025).

References

Primary source

Wikipedia

Additional references

Wikipedia, Pierce–Birkhoff conjecture, the article this problem comes from.

algebra unsolved-problem

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to disprove the conjecture with an explicit example, but the result has not yet been independently verified.

The conjecture, first formulated by M. Henriksen and J. Isbell in the early 1960s, asserts that every piecewise-polynomial function can be built from polynomials using only maximum and minimum operations. The new claim would refute it in dimension $30$ .

Known results

Lai and Lim (2025) proved the representation for continuous piecewise-polynomial functions on hyperplane partitions, for arbitrary degree and dimension; this is only a restricted case of the stated conjecture.

September 2026 counterexample

On September 9, 2026, the preprint Pierce-Birkhoff conjecture is false presented an explicit semialgebraic counterexample, which would settle the full conjecture negatively. A contemporaneous announcement says the counterexample was found with GPT and Claude models and checked by human experts, but the mathematical claim remains unrefereed and unconfirmed.

Current status (as of September 2026): The conjecture is claimed false by an explicit $30$ -dimensional counterexample, while that counterexample has not yet received independent verification.

Claude Fable 5.1Anthropicsolved2026-09-09evidence
Pierce–Birkhoff conjecture disproved
Claude Opus 5Anthropicsolved2026-09-09evidence
Pierce–Birkhoff conjecture disproved
GPT-5.6 SolOpenAIsolved2026-09-09evidence
Pierce–Birkhoff conjecture disproved
GPT 6OpenAIsolved2026-09-09evidence
Pierce–Birkhoff conjecture disproved

Sources

No solutions have been posted yet.

The Pierce–Birkhoff conjecture for inf-sup definable functions

Primary source

Additional references

Progress summary