Model theory of analytic functions: some historical comments (Q2915887)

This paper gives an overview of the history of the model-theoretic treatment of analytic functions. More precisely, it mainly considers the model theory of topological fields expanded by such functions. The biggest part concerns the field \(\mathbb{R}\), but \(\mathbb{C}\) and some valued fields are mentioned, too. Types of results that are considered in various contexts are decidability, model completeness, quantifier elimination, elimination of imaginaries, and -- over \(\mathbb{R}\) -- o-minimality. To give a more precise impression of the contents, let me simply list some keywords.NEWLINENEWLINEThe paper considers expansions of \(\mathbb{R}\) by restricted analytic functions, by the exponential map, by Pfaffian chains, and by a Rolle leaf. Types of subsets of \(\mathbb{R}^n\) include semi-algebraic sets and (globally) semi- and sub-analytic sets.NEWLINENEWLINEOver \(\mathbb{C}\), the paper mentions the model theory of compact complex varieties and results concerning \(\mathbb{C}\) expanded by the exponential map.NEWLINENEWLINEThe valued fields that are mentioned are \(\mathbb{Q}_p\), algebraically closed fields and fields of power series (again, possibly expanded by restricted analytic functions).NEWLINENEWLINEOther things that are mentioned are Schanuel's conjecture (and its relation to various of the above problems), differentiably closed fields, Hardy fields, fields of germs at \(\infty\), and a model-theoretic approach to Berkovich spaces.