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Preprint Number 483
483. François Lucas, Daniel Schaub, Mark Spivakovsky
On the Pierce-Birkhoff Conjecture
Submission date: 27 July 2012.
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 \alpha,\beta\in\sper\ A and their separating ideal <\alpha,\beta>; we refer to this statement as the Local Pierce-Birkhoff conjecture at \alpha,\beta. In this paper, for each pair (\alpha,\beta) with ht(<\alpha,\beta>)=\dim A, we define a natural number, called complexity of (\alpha,\beta). Complexity 0 corresponds to the case when one of the points \alpha,\beta 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(<\alpha,\beta>) 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(<\alpha,\beta>) 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(<\alpha,\beta>)=3, the pair (\alpha,\beta) is of complexity 1 and A is excellent with residue field the field of real numbers.
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