Some structures interpretable in the ring of continuous semi-algebraic functions on the line.

Laura Phillips (Manchester)

Frank Adams 1,

For a real closed field \(R\), let \(C^\mathrm{sa}(R^n, R)\) be the ring of continuous, semi-algebraic functions \(R^n\to R\). The first order \(\mathcal{L}_\mathrm{ring}\)-theory of \(C^\mathrm{sa}(R^n,R)\) is known to be undecidable when \(n\geq 2\). I will give some background and describe some of the model-theoretic work undertaken on the rings \(C^\mathrm{sa}(R^n,R)\). I will then describe some decidability results relevant to the 1-dimensional case. In particular I'll discuss work on the theory of free modules over this ring, and Astier's treatment of the ring of semi-algebraic functions \(R\to R\) (i.e. no continuity requirement). If there is time I will talk about some work in progress concerning the status of Astier's result when one demands continuity of functions at some fixed point of \(R\).

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