arXiv subjects

Françoise Point

Publications and source records attributed to Françoise Point.

12 recordsLinked to original sources

arXiv2025

On definable groups in dp-minimal topological fields equipped with a generic derivation

Let $T$ be a complete, model-complete, geometric dp-minimal $\mathcal{L}$-theory of topological fields of characteristic $0$ and let $T(\partial)$ be the theory of expansions of models of $T$ by a derivation $\partial$. We assume that $T(\partial)$ has a model-companion $T_{\partial}$. Let $\Gamma$ be a finite-dimensional $\mathcal{L}_\partial$-definable group in a model of $T_\partial$. Then we show that $\Gamma$ densely and definably embeds in an $\mathcal{L}$-definable group $G$. Further, using a $C^1$-cell decomposition result, we show that $\Gamma$ densely and definably embeds in a definable $D$-group, generalizing the classical construction of Buium of algebraic $D$-groups and extending for that class of fields, results obtained in arXiv:2208.08293, arXiv:2305.16747.

math.LO↗

arXiv2025

On groups definable in geometric fields with generic derivations

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

math.LO↗

arXiv2023

Dense pairs of rings

Outside of the framework of geometric theories, we exhibit complete, respectively model-complete theories of rings whose corresponding theory of pairs is complete, respectively model-complete, using transfer results proven in the seventies for boolean products of structures. It includes certain boolean products of pairs of dp-minimal fields of characteristic $0$. We also show, as in the case of pairs of fields, how it fits in the framework of differential rings.

math.LO↗

arXiv2023

A decidable expansion of $(\Gamma,+,F)$ with the independence property

Let $(\Gamma,+,F)$ be a finitely generated $\mathbb Z[F]$-module where $F$ is an injective endomorphism of the abelian group $\Gamma$. We restrict ourselves to a finite automa presentable subclass, introduced by J. Bell and R. Moosa in "F-sets and finite automata. J. Th\'eor. Nombres Bordeaux 31 (2019), no. 1, 101-130" and define an expansion containing the $\mathcal F$-sets defined by R. Moosa and T. Scanlon in "Am. J. Math. 126 (2004), no. 3, p. 473-522", where every automatic subset is definable.

math.LO↗

arXiv2022

Topological fields with a generic derivation

We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_δ$-extension $T_δ^*$ by a generic derivation $δ$. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has $\mathcal{L}$-open core (i.e., every $\mathcal{L}_δ$-definable open set is $\mathcal{L}$-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of $T$ such as relative elimination of field sort quantifiers, NIP and distality also transfer to $T_δ^*$. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of $p$-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

math.LO↗

arXiv2020

On expansions of $(\mathbf{Z},+,0)$

Call a (strictly increasing) sequence $(r_{n})$ of natural numbers \emph{regular} if it satisfies the following condition: $r_{n+1}/r_{n}\toθ\in\mathbb{R}^{>1}\cup\{\infty\}$ and, if $θ$ is algebraic, then $(r_{n})$ satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of $θ$. Our main result states that $(\mathbb{Z},+,0,R)$ is superstable whenever $R$ is enumerated by a regular sequence. We give two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. We also show that $(\mathbb{Z},+,0,<,R)$ is NIP whenever $R$ is enumerated by a regular sequence that is ultimately periodic modulo $m$ for all $m>1$.

math.LO↗

arXiv2018

Bézout domains and lattice-valued modules

Let B be a commutative Bézout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes of modules over the localizations of B by the maximal ideals of B, and on the other hand, of the constructible subsets of MSpec(B). When B has good factorization, it allows us to derive decidability results for the class B-modules, in particular when B is the ring of algebraic integers or its intersection with real numbers or p-adic numbers.

math.LO↗

arXiv2017

Definable groups in topological differential fields

For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $\mathcal L\cup\{D\}$-definable sets and their $\mathcal L$-reducts, where $\mathcal L$ is a relational expansion of the field language and $D$ a symbol for a derivation. This enables us to associate with an $\mathcal L\cup\{D\}$-definable group in models of such theories, a local $\mathcal L$-definable group. As a byproduct, we show that in closed ordered differential fields, one has the descending chain condition on centralisers.

math.LO↗

arXiv2017

Fractional Parts of Dense Additive Subgroups of Real Numbers

Given a dense additive subgroup $G$ of $\mathbb R$ containing $\mathbb Z$, we consider its intersection $\mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $\mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a "standard" part and two ordered semigroups of infinitely small and infinitely large elements.

math.LO↗

arXiv2016

On differential Galois groups of strongly normal extensions

We give a detailed proof of Kolchin's results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and his co-authors which were written under the assumption that the field of constant is algebraically closed. In the present setting, which encompasses the cases of ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in the theory of closed ordered fields, we establish a relative Galois correspondence for definable subgroups of the group of differential order automorphisms.

math.LO↗

arXiv2014

Separably closed fields and contractive Ore modules

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

math.LO↗

arXiv2012

Alternatives for pseudofinite groups

The famous Tits' alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an $\aleph_{0}$-saturated pseudofinite group either contains a subsemigroup of rank $2$ or is nilpotent-by-(uniformly locally finite). We call a class of finite groups $G$ weakly of bounded rank if the radical $rad(G)$ has a bounded Prüfer rank and the index of the sockel of $G/rad(G)$ is bounded. We show that an $\aleph_{0}$-saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between this kind of alternatives and amenability.

math.GR↗