2012/07/27 by Lucas, François, Schaub, Daniel, Spivakovsky, Mark · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1207.6463
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 Ais equivalent to a statement about an arbitrary pair of points α,β∈\sper A and their separating ideal ; we refer to this statement as the Local Pierce-Birkhoff conjecture at α,β. In this paper, for each pair (α,β) with ht()=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() 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() 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()=3, the pair (α,β) is of complexity 1 and A is excellent with residue field the field of real numbers.