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Axiomatization of local-global principles for pp-formulas in spaces of orderings
 

Summary: Axiomatization of local-global principles for
pp-formulas in spaces of orderings
V. Astier, M. Tressl
Two important results in quadratic form theory, Pfister's local-global
principle and the isotropy theorem (see [5]), can be stated more generally
for spaces of orderings (an abstract version of real spectras of formally real
fields), for which they are expressed as local-global principles:
A property of quadratic forms (expressed as a so-called positive-primitive
formula) holds if and only if it holds locally (at every single ordering for Pfis-
ter's local-global principle, at every finite subspace for the isotropy theorem).
In his paper [7], Marshall introduces a much broader local-global principle
that could be satisfied by spaces of orderings, and showed how several im-
portant questions about quadratic forms and real algebraic geometry would
follow from it. He asks, for a space of orderings (X, G) (the unexplained
terminology will be introduced later):
"Is it true that any positive-primitive formula (Żg) with
parameters Żg in G which holds in every finite subspace
of (X, G) necessarily holds in (X, G)?"
(LG)
No counter-examples are known. In this paper we work in the theory of

  

Source: Astier, Vincent - Department of Mathematics, University College Dublin

 

Collections: Mathematics