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$isequivalenttoastatementaboutanarbitrarypairofpoints$α,β\in\sper\ A$andtheirseparatingideal$<α,β>$;werefertothisstatementastheLocalPierce-Birkhoffconjectureat$α,β$.Inthispaper,foreachpair$(α,β)$with$ht(<α,β>)=\dim A$,wedefineanaturalnumber,calledcomplexityof$(α,β)$.Complexity0correspondstothecasewhenoneofthepoints$α,β$ismonomial;thiscasewasalreadysettledinalldimensionsinaprecedingpaper.Hereweintroduceanewconjecture,calledtheStrongConnectednessconjecture,andprovethatthestrongconnectednessconjectureindimensionn-1impliestheconnectednessconjectureindimensionninthecasewhen$ht(<α,β>)$islessthann-1.WeprovetheStrongConnectednessconjectureindimension2,whichgivestheConnectednessandthePierce--Birkhoffconjecturesinanydimensioninthecasewhen$ht(<α,β>)$lessthan2.Finally,weprovetheConnectedness(andhencealsothePierce--Birkhoff)conjectureinthecasewhendimensionofAisequalto$ht(<α,β>)=3$,thepair$(α,β)$isofcomplexity1and$A$ is excellent with residue field the field of real numbers.