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

#### Laura Phillips (Manchester)

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 $$R$$.