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The definable connectedness conjecture for regular rings

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Let AA be a regular ring, let α,β∈Sper⁡A\alpha,\beta\in\operatorname{Sper} A, and let g1,…,gsg_1,\dots,g_s be a finite collection of elements of AA not belonging to ⟨α,β⟩\langle\alpha,\beta\rangle. A subset CC of Sper⁡A\operatorname{Sper} A is definably connected if it is not a union of two non-empty disjoint constructible subsets that are relatively closed for the spectral topology. Definable connectedness conjecture. There exists a definably connected set C⊂Sper⁡AC\subset\operatorname{Sper} A such that α,β∈C\alpha,\beta\in C and

C∩{gi=0}=∅C\cap\{g_i=0\}=\emptyset

for i∈{1,…,s}i\in\{1,\dots,s\}. Equivalently, α\alpha and β\beta belong to the same definably connected component of

Sper⁡A∖{g1⋯gs=0}.\operatorname{Sper} A\setminus\{g_1\cdots g_s=0\}.

This conjecture refines the preceding connectedness formulation by requiring definable connectedness in the spectral topology; its status is not resolved in the supplied source.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The definable connectedness conjecture for regular rings

    Let AA be a regular ring and let α,β∈Sper⁡A\alpha,\beta\in\operatorname{Sper} A. A subset of Sper⁡A\operatorname{Sper} A is definably connected if it is not the union of two nonempty disjoint constructible subsets that are relatively closed in the spectral topology. Say that AA has the definable connectedness property at α,β\alpha,\beta if, for every finite collection g1,…,gs∈A∖⟨α,β⟩g_1,\dots,g_s\in A\setminus\langle\alpha,\beta\rangle, there is a definably connected set C⊂Sper⁡AC\subset\operatorname{Sper} A containing α,β\alpha,\beta and avoiding every zero set {gi=0}\{g_i=0\}.

    Definable connectedness conjecture. A regular ring AA satisfies the definable connectedness property at every pair α,β∈Sper⁡A\alpha,\beta\in\operatorname{Sper} A.

    The definable version is designed to imply the Pierce–Birkhoff conjecture by the same argument as ordinary connectedness. Its general validity is left open in the paper.

    source: François Lucas, Daniel Schaub and Mark Spivakovsky, “On the Pierce-Birkhoff Conjecture”, arXiv:1207.6463 (2012).

References

Primary source

François Lucas, James Madden, Daniel Schaub and Mark Spivakovsky, “Approximate roots of a valuation and the Pierce-Birkhoff Conjecture”, arXiv:1003.1188 (2012).

pierce-birkhoffreal-algebraic-geometryreal-spectrum
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