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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 already settled in all dimensions in a preceding paper. Here 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 $ht(<α,β>)$ is less than 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 $ht(<α,β>)$ less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to $ht(<α,β>)=3$, the pair $(α,β)$ is of complexity 1 and $A$ is excellent with residue field the field of real numbers.