Definable types in algebraically closed valued fields (Q2793905)

One way to characterize stability is that a theory is stable if and only if every type over every model is definable. There are, however, individual models of unstable theories in which types are definable. For example, all types over the real field are definable [\textit{L. van den Dries} in: Logic colloquium '84. Proceedings of the Colloquium held in Manchester, U.K., July 1984. Amsterdam etc.: North-Holland. 59--90 (1986; Zbl 0616.03018)], as are types over the \(p\)-adics [\textit{F. Delon}, Proc. Amer. Math. Soc. 106, 193--198 (1989; Zbl 0726.03028)]. One might then ask which other models of unstable theories have this property.NEWLINENEWLINEFor models \(M\prec N\), the authors write \(T_n(M,N)\) to mean that \(\mathrm{tp}(a/M)\) is definable for all \(a\in N^n\). Using this notation, a theory is stable if and only if \(\bigwedge_{n\geq 1} T_n(M,N)\) for all models \(M\prec N\). Similarly, van den Dries's result [loc. cit.] shows that even though the theory of real closed fields is unstable, if we fix \(M=\mathbb R\) then \(\bigwedge_{n\geq 1} T_n(\mathbb R,N)\) for all \(N\succ \mathbb R\).NEWLINENEWLINE\textit{D. Marker} and \textit{C. I. Steinhorn} [J. Symb. Log. 59, No. 1, 185--198 (1994; Zbl 0801.03026)] showed that for o-minimal structures, it suffices to check 1-types; that is, for o-minimal structures \(M\prec N\), \(T_1(M,N)\) implies \(\bigwedge_{n\geq 1} T_n(M,N)\). This result does not hold in certain generalizations of o-minimal structures, including weakly o-minimal structures [\textit{B. S. Baizhanov}, Sib. Adv. Math. 16, No. 2, 2--33 (2006; Zbl 1249.03070)] and NIP structures [\textit{A. Chernikov} and \textit{P. Simon}, Trans. Amer. Math. Soc., 367, 5217--5235 (2015; Zbl 1388.03035)]. In this paper, the authors investigate the case of algebraically closed non-trivially valued fields (ACVF), and show that the Marker-Steinhorn result [loc. cit.] does not hold in ACVF.NEWLINENEWLINEThe second section of the paper focuses on the proof of the following:NEWLINENEWLINE Theorem. For models \(K\prec L\) of ACVF, \(\bigwedge_{n\geq 1} T_n(K,L)\) holds if and only if the valued field extension is separated and \(T_1(vK,vL)\) holds (where \(vK\) and \(vL\) are the value groups of \(K\) and \(L\)).NEWLINENEWLINE The third section uses the results of the second section to build models \(K\prec L\) of ACVF in which \(T_n(K,L)\) holds but \(T_{n+1}(K,L)\) does not (for every \(n\geq 1\)). These models provide the counterexample to the Marker-Steinhorn result [loc. cit.] in ACVF.NEWLINENEWLINEThe results in the paper are well written and easy to understand, although the overall structure can be difficult to follow at times. The paper uses a number of results about the structure of models of ACVF; for this reason, the paper will be easier to read for someone familiar with the general model theory of valued fields, and ACVF in particular.