Perfect pseudo-algebraically closed fields are algebraically bounded.
Algebraic boundedness for theories \(T\) of fields is a concept that was introduced by L. Van den Dries, regarding, for instance, algebraically closed fields, real closed fields and \(p\)-adic fields, and ensuring a well-behaved notion of dimension on definable sets \(S\) of models of \(T\) (essentially coinciding with the algebraic dimension of the Zariski closure of \(S\)). In fact, \(T\) is said to be algebraically bounded if and only if for every formula \(\varphi(\overline{x}, \, y)\) there are finitely many polynomials \(f_j (\overline{x}, \, y)\) with coefficients from \({\mathbb Z}\) such that, for any model \(K\) of \(T\) and \(t\)-tuple \(\overline{a}\) in \(K\), there is some \(j\) for which \(f_j(\overline{a}, \, y)\) is not identically \(0\) in \(K\) and \(\varphi(\overline{a}, \, K)\) is just contained in the solution set of \(f_j(\overline{a}, \, y) = 0\) in \(K\). So, algebraic boundedness is a strong uniformity notion on finite definable sets. The authors consider here pseudo-algebraically closed fields (i.e., fields \(K\) where every absolutely irreducible variety over \(K\) has a \(K\)-rational point), so a quite intriguing framework in Model Theory, and show that every theory of perfect pseudo-algebraically closed fields is algebraically bounded. This extends a previous result of Jarden on perfect Frobenius fields.
scientific article; zbMATH DE number 2037104