2018/02/26 by F Lucas, Lucas, F, Daniel Schaub +3
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1802.09389
openalex publication_date 2018/02/26 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28
This paper contains a partial result on the Pierce--Birkhoff conjecture on piece-wise polynomial functions defined by a finite collection f 1,. .., f r of polynomials. In the nineteen eighties, generalizing the problem from the polynomial ring to an artibtrary ring Σ, J. Madden proved that the Pierce--Birkhoff conjecture for Σ is equivalent to a statement about an arbitrary pair of points α, β ∈ Sper Σ and their separating ideal < α, β >, we refer to this statement as the local Pierce-Birkhoff conjecture at α, β. In [8] we introduced a slightly stronger conjecture, also stated for a pair of points α, β ∈ Sper Σ and the separating ideal < α, β >, called the Connectedness conjecture, about a finite collection of elements f 1, . . ., fr ⊂ Σ. In the paper [10] we introduced a new conjecture, called the Strong Connectednessconjecture, and proved that the Strong Connectedness conjecture in dimension n--1 implies the Strong Connectedness conjecture in dimension n in the case when ht(< α, β >) ≤ n -- 1.The Pierce-Birkhoff Conjecture for r = 2 is equivalent to the Connectedness Conjecture for r = 1, this conjecture is called the Separation Conjecture. The Strong Connectedness Conjecture for r = 1 is called the Strong Separation Conjecture. In the present paper, we fix a polynomial f ∈ R[x, z] where R is a real closed field and x = (x1, . . ., xn), z are n + 1 independent variables. We define the notion of two points α, β ∈ Sper R[x, z] being in good position with respect to f. The main result of this paper is a proof of the Strong Separation Conjecture in the case when α and β are in good position with respect to f.