On the Pierce–Birkhoff conjecture

Abstract

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∈SperA 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∈SperA and the separating ideal <α,β>, called the Connectedness conjecture. 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 settled in all dimensions in [8]. In the present paper we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n−1 implies the connectedness conjecture in dimension n in the case when ht(<α,β>)≤n−1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce–Birkhoff conjectures in any dimension in the case when ht(<α,β>)≤2. Finally, we prove the Connectedness (and hence also the Pierce–Birkhoff) conjecture in the case when dim⁡A=ht(<α,β>)=3, the pair (α,β) is of complexity 1 and A is excellent with residue field R.