A version of \(p\)-adic minimality (Q2892680)

One of the fundamental features of real closed fields is that they are o-minimal. The present article introduces \(\mathcal L_M\)-minimality, an analogue of this for \(p\)-adically closed fields (such as \(\mathbb Q_p\)). Such an analogue already exists, namely the notion of \(P\)-minimality introduced by \textit{D. Haskell} and \textit{D. Macpherson} [J. Symb. Log. 62, No. 4, 1075--1092 (1997; Zbl 0894.03015)]. However, \(P\)-minimality requires the language to contain the full field language, which seems unsatisfactory given that o-minimality can be defined using only the order. By contrast, \(\mathcal{L}_M\)-minimality needs only a very weak language \(\mathcal{L}_M\).NEWLINENEWLINELike o-minimality, the notion of \(\mathcal{L}_M\)-minimality is a special case of the general notion of \(\mathcal L_0\)-minimality introduced by \textit{D. Macpherson} and \textit{C. Steinhorn} [Ann. Pure Appl. Logic 79, No. 2, 165--209 (1996; Zbl 0858.03039)]: a structure is \(\mathcal{L}_M\)-minimal if every definable subset of the line is \(\mathcal{L}_M\)-definable. Any valued field with finite residue field and whose value group is elementarily equivalent to \(\mathbb Z\) naturally carries \(\mathcal{L}_M\)-structure. The authors prove that, using this, \(p\)-adically closed fields are \(\mathcal{L}_M\)-minimal. On finite field extensions of \(\mathbb Q_p\), one even has \(\mathcal{L}_M\)-minimality for the subanalytic language.NEWLINENEWLINEOther results are that \(\mathcal{L}_M\)-minimality implies cell decomposition and there is a quantifier elimination result in a suitable definitional expansion of \(\mathcal{L}_M\).NEWLINENEWLINETo obtain a general notion of \(\mathcal{L}_M\)-minimality, one would need an axiomatic description of those \(\mathcal{L}_M\)-structures one is interested in (in analogy to the requirement that, in o-minimal structures, \(<\) defines a dense linear order). As the authors remark, such an axiomatic description is still missing.