Abstract
Let R be a real closed field, and \({X\subseteq R^m}\) semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions \({X\to R}\) definable with parameters in R. As a tool we introduce (pre)-piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of syzygies of finitely generated submodules of free modules. We give an explicit method to determine the Baur–Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients.
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Phillips, L.R. Positive primitive formulae of modules over rings of semi-algebraic functions on a curve. Arch. Math. Logic 54, 587–614 (2015). https://doi.org/10.1007/s00153-015-0429-8
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DOI: https://doi.org/10.1007/s00153-015-0429-8
Keywords
- Rings of continuous semi-algebraic functions
- Semi-algebraic vector bundles
- Piecewise vector bundles
- p.p.-Formula
- Model theory of modules
- Semi-algebraic geometry