Positive primitive formulae of modules over rings of semi-algebraic functions on a curve

Abstract

Let R be a real closed field, and $${X\subseteq R^m}$$X⊆Rm semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions $${X\to R}$$X?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.