On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approxim…

Pierce-Birkhoff 猜想对样条函数成立

Pierce-Birkhoff conjecture is true for splines

本文证明了 Pierce-Birkhoff 猜想对样条函数成立，即连续分片多项式可表示为多项式的有限格组合，并进一步给出了有效界。

On the Pierce-Birkhoff Conjecture

This paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points $α,β\in\sper\ A$ and their separating ideal $<α,β>$; we refer to this statement as the Local Pierce-Birkhoff conjecture at $α,β$. In this paper, for each pair $(α,β)$ with $ht(<α,β>)=\dim A$, we define a natural number, called complexity of $(α,β)$. Complexity 0 corresponds to the case when one of the points $α,β$ is monomial; this case was alrea…

Pierce-Birkhoff 猜想是错误的

Pierce-Birkhoff conjecture is false

本文通过多智能体多模型协作，构造了一个连续逐段二次函数作为反例，证明 Pierce-Birkhoff 猜想不成立。

Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials

Let R denote the reals, and let h: R^n --> R be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup_i inf_j f_{ij}, for some finite collection of polynomials f_{ij} in R[x_1,...,x_n]. (A simple example is h(x_1) = |x_1| = sup{x_1, -x_1}.) In 1984, L. Mahe and, independently, G. Efroymson, proved this for n < 3; it remains open for n > 2. In this paper we prove an analogous result for "generalized polynomials" (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not …

A connectedness theorem for real spectra of polynomial rings

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on R^n can be obtained from the polynomial ring R[x_1,...,x_n] by iterating the operations of maximum and minimum. The purpose of this paper is twofold. First, we state a new conjecture, called the Connectedness conjecture, which asserts the existence of connected sets in the real spectrum of R[x_1,...,x_n] satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce-Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum …