arXiv subjects
Publications and source records attributed to Ya'acov Peterzil.
We show that every definable function in a weakly o-minimal expansion of a field $K$ is generically differentiable, where the derivative takes values in the Cauchy completion of $K$, i.e. it is a definable nonvaluational cut. As a consequence, every definable field is definably isomorphic to either $K$ or its algebraic closure.
For an $\aleph_0$-saturated weakly o-minimal expansion of an ordered group $\mathcal{M}$, it is shown that $\mathcal{M}^{\text{eq}}$ has uniform finiteness if and only if the collection of definable convex subgroups of $\mathcal{M}$ has uniform finiteness. If $\mathcal{M}$ expands an ordered field, then considering definable convex valuation subrings is sufficient. The results use a criterion of Johnson for uniform finiteness in $\mathcal{M}^{\text{eq}}$, [8]. In addition, it is shown that uniform finiteness in $\mathcal{M}^{\text{eq}}$ may fail for weakly o-minimal expansions of fields.
We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $Γ$, and the closed $0$-balls $K/\mathcal{O}$.
We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $ν(G)$, generated by the four infinitesimal subgroups $ν_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $Γ$ and $K/\mathcal{O}$. To show that $ν(G)$ is type-definable, we show that the resulting subgroups $ν_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $ν(G)$. Among others, we show that $ν(G_1\times G_2)=ν(G_1)\times ν(G_2)$ and that if $G_1\le G$ is a definable subgroup then $ν(G_1)$ is relatively definable in $ν(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(ν(G))$ and elimination of imaginaries.
Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $Γ\subseteq G$, we study the possible Hausdorff limits of $π(\mathcal F)$ in $G/Γ$ as $t$ tends to $\infty$ (here $π:G\to G/Γ$ is the canonical projection). Towards a solution, we associate to $\mathcal F$ finitely many real algebraic subgroups $L\subseteq G$, and, uniformly in $Γ$, determine if the only Hausdorff limit at $\infty$ is $G/Γ$, depending on whether $L^Γ=G$ or not. The special case of polynomial dilations of a definable set is treated in details.
We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field $K$, its residue field $\mathbf{k}$ (when infinite), its value group $Γ$, or $K/\mathcal{O}$, where $\mathcal{O}$ is the valuation ring. Our work uses and extends techniques developed in [11] to circumvent elimination of imaginaries.
We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. We generalize Buium's notion of an algebraic D-group to L-definable D-groups, namely (G,s), where G is a L-definable group in a model of T, and s is an L-definable group section into the prolongation of G. Our main theorem says that every definable group of finite dimension in a model of the theory is definably isomorphic to the ``sharp'' points of an L-definable D-group. We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic zero.
A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in: 1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic $0$); and 2) $o$-minimal expansions of groups. Our methods provide explicit bounds on the power saving exponent in the non-group case. Ingredients of the proof include: a higher arity generalization of the abelian group configuration theorem in stable structures, along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups; and Zarankiewicz-style bounds for hypergraphs definable in distal structures.
We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.
Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that $G/H$ is definably isomorphic to a $K$-linear group. The result remains true in models of $\mathrm{Th}(\mathbb{Q}_p^{an})$.
Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct $\mathcal M$ of a linearly ordered structure $\langle R;<,\cdots\rangle $ is called \emph{strongly bounded} if every $\mathcal M$-definable subset of $R$ is either bounded or co-bounded in $R$. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.
Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\mathcal{F}$ an infinite field interpretable in $\mathcal{K}$. Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii) power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then $\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\mathbb{Q}_p$, answering a question of Pillay's. Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if $\mathcal{K}$ is an infinite dp-minimal pure field then every field definable in $\mathcal{K}$ is definably isomorphic to a finite extension of $K$. The proof avoids elimination of imaginaries in $\mathcal{K}$ replacing it with a reduction of the problem to certain distinguished quotients of $K$.
Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The same result holds when $K$ is a model of $T$, for $T$ an o-minimal power bounded expansion of a real closed field, and $O$ is a $T$-convex subring. The proof is direct and does not make use of known results about elimination of imaginaries in valued fields.
Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $Γ$ be a lattice in $G$, with $π:G\to G/Γ$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $π(X)$ in the compact nilmanifold $G/Γ$. Our theorem describes $\mathrm{cl}(π(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $Γ$. We also prove an equidistribution result in the case of curves.
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.
We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field where definable functions are generically differentiable.
The group configuration in o-minimal structures gives rise, just like in the stable case, to a transitive action of a type-definable group on a partial type. Because $acl=dcl$ the o-minimal proof is significantly simpler than Hrushovski's original argument. Several equivalent versions, which are more suitable to the o-minimal setting, are formulated, in functional language and also in terms of a certain $4$-ary relation. In addition, the following question is considered: Can every definably connected type-definable group be definably embedded into a definable group of the same dimension? Two simple cases with a positive answer are given.
We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.