On the strong separation conjecture

Abstract

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 ; 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 , 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 Connectedness conjecture, and proved that the Strong Connectedness conjecture in dimension n−1 implies the Strong Connectedness conjecture in dimension n in the case when ht( ) ≤ 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.