- Stabilizers, Measures and IP-sets, with Daniel Palacín,
to appear in Research Trends in Contemporary Logic, Ed: M. Fitting et al. Arxiv
We provide elementary
model-theoretic proofs to some existing results on sumset phenomena
and IP sets, motivated by Hrushovski's work on the stabilizer
- A model-theoretic note on the Freiman-Ruzsa theorem, with
Daniel Palacín and Julia Wolf, Sel. Math. New Ser. 27 (2021), DOI: https://doi.org/10.1007/s00029-021-00676-9. Arxiv
A non-quantitative version
of the Freiman-Ruzsa theorem is obtained for finite stable sets with
small tripling in arbitrary groups, as well as for (finite) weakly
normal subsets in abelian groups.
- Trois couleurs: A new non-equational theory, with Martin Ziegler,
Fund. Math. 254 (2021), 313--333. DOI: https://doi.org/10.4064/fm953-9-2020. Arxiv
A first-order theory is equational if every definable set is a
Boolean combination of instances of equations, that is, of formulae
such that the family of finite intersections of instances has the
descending chain condition. Equationality is a strengthening of
stability yet so far only two examples of non-equational stable
theories are known. We construct non-equational ω-stable theories by
a suitable colouring of the free pseudospace, based on Hrushovski
and Srour's original example.
- Open core and small groups in dense pairs of topological structures, with Elías Baro,
J. Pure & Appl. Logic 172 (2021), DOI:
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.
- Equational theories of fields, with Martin Ziegler, J. Symbolic Logic 85 (2020), 828 -- 851.
1702.05735 or shorter version.
A complete first-order
theory is equational if every definable set is a Boolean combination
of instances of equations, that is, of formulae such that the family
of finite intersections of instances has the descending chain
condition. Equationality is a strengthening of stability. We show
the equationality of three theories of fields: the theory of proper
extension of algebraically closed fields of some fixed
characteristic, the theory of differentially closed fields in
positive characteristic and the theory of separably closed fields of
infinite imperfection degree. As a by-product, we obtain relative
elimination of imaginaries for the the theory of separably closed
fields of infinite imperfection degree, after adding sorts for the
canonical parameters of equations.
- Un critère simple, with Thomas Blossier,
Notre Dame J. Formal Logic 60 (2019), 639--663.
In this short note, we
mimic the proof of the simplicity of the theory ACFA of generic
difference fields in order to provide a criterion, valid for certain
theories of pure fields and fields equipped with operators, which
shows that a complete theory is simple whenever its definable and
algebraic closures are controlled by an underlying stable
- Ample pairs, with Enrique Casanovas and Daniel Palacín,
Fund. Math. 247 (2019), 37--48.
We show that the
ample degree of a stable theory with trivial forking is preserved
when we consider the corresponding theory of belles paires, if it
exists. This result also applies to the theory of H-structures of
a trivial theory of rank 1.
- Sur les automorphismes bornés de corps munis
d'opérateurs, with Thomas Blossier and Charlotte Hardouin, Math. Research Letters 24
(2017), 955--978. Arxiv
We give an alternative
proof, valid in all characteristics, of a result of Lascar
characterising the bounded automorphisms of an algebraically closed
field. We generalise this method to various fields equipped with
- A Model Theoretic Study of Right-Angled Buildings,
with Andreas Baudisch and Martin Ziegler, J. Eur. Math. Soc. 19 (2017),
We study the
model theory of countable right-angled buildings with infinite
residues. For every Coxeter graph we obtain a complete theory with a
natural axiomatisation, which is ω-stable and
equational. Furthermore, we provide sharp lower and upper bounds for
its degree of ampleness, computed exclusively in terms of the
associated Coxeter graph. This generalises and provides an
alternative treatment of the free pseudospace.
- A la recherche du tore perdu, with T. Blossier et F. O. Wagner, J. Symbolic
Logic 81 (2016), 1--31, HAL-00758982.
We classify the
groups definable in the coloured fields obtained by Hrushovski
amalgamation. A group definable in the bad green field is isogenous
to the quotient of a subgroup of an algebraic group by a Cartesian
power of the group of green elements. A definable subgroup of an
algebraic group in the green or red field is an extension of a
Cartesian power of the subgroup of coloured elements by an algebraic
group. In particular, a simple group in a coloured field is
- Géométries relatives, with T. Blossier et F. O. Wagner,
J. Eur. Math. Soc. 17 (2015), 229--258, HAL-00514393.
We start an
analysis of geometric properties of a structure relative to a
reduct. In particular, we look at definability of groups and fields
in this context. In the relatively one-based case, every definable
group is isogenous to a subgroup of a product of groups definable in
the reducts. In the relatively CM-trivial case, which contains
certain Hrushovski amalgamations (the fusion of two strongly minimal
sets or the expansions of a field by a predicate), every definable
group allows a homomorphism with virtually central kernel into a
product of groups definable in the reducts.
- De beaux groupes, with T. Blossier, Confl. Math. 6
(2014), 3--13, HAL-00837759.
In this short
paper, we will provide a characterisation of interpretable groups in
a beautiful pair (K, E) of algebraically closed fields: every
interpretable group is, up to isogeny, the extension of the subgroup
of E-rational points of an algebraic group by an interpretable group
which is the quotient of an algebraic group by the E-rational points
of an algebraic subgroup.
- Ample Hierarchy, with
Andreas Baudisch and Martin Ziegler, Fund. Math. 224 (2014), 97-153, HAL-00863214.
The ample hierarchy of geometries of stables theories is strict. We generalise
the construction of the free pseudospace to higher dimensions and show that the n-dimensional free
pseudospace is ω-stable n-ample yet not (n+1)-ample. In particular, the free pseudospace is
not 3-ample. A thorough study of forking is conducted and an explicit description of canonical
bases is exhibited.
- On variants of CM-triviality, with T. Blossier et F. O. Wagner, Fund. Math. 219 (2012), 253--262,
We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial
types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which
generalizes one-basedness. We show that, under this condition, a stable field is internal to the
family, and a group of finite Lascar rank has a normal nilpotent subgroup such that the quotient is
almost internal to the family.
- Supersimplicity and quadratic extensions, with F. O. Wagner, Archive for Math. Logic 48 (2009), 55--61,
Elliptic curves over a supersimple field with exactly one extension of degree 2 have
s-generic rational points.
- Sur les collapses de corps différentiels colorés en caractéristique nulle
décrits par Poizat à l'aide des amalgames à la Hrushovski, with T. Blossier, J. Inst. Math. Jussieu 8 (2009),
445 -- 464, HAL-00261500.
We collapse Poizat's red fields in characteristic 0 to obtain a differentially closed field
of rank ω ⋅ 2 equipped with a definable additive subgroup of commensurable rank. We
obtain by using the logarithmic derivative a green multiplicative subgroup which cannot
be of finite rank.
- Die böse Farbe, with
Andreas Baudisch, Martin Hils and F. O. Wagner,
J. Inst. Math. Jussieu 8 (2009), 415--443,
Modnet Preprint 12.
We construct a bad field in characteristic zero. That is, an algebraically closed field with a notion
of dimension analogous to Zariski dimension, equipped with an infinite proper multiplicative subgroup
of dimension one, such that the field itself has dimension 2. This answers a longstanding open question
- Red fields, with
Andreas Baudisch and Martin Ziegler,
J. Symb. Logic 72 (2007), 207--225,
Modnet Preprint 13.
We apply Hrushovski-Fraïssé's amalgamation procedure to
obtain a theory of fields of prime characteristic of Morley rank
2 equipped with a definable additive subgroup of rank 1.
- Hrushovski's Fusion, with
Andreas Baudisch and Martin Ziegler,
Algebra, Logic, Set Theory, Festschrift für Ulrich Felgner zum 65 Geburtstag., 2007,
Frieder Haug, Benedikt Löwe, Torsten Schatz (eds.), Studies in Logic, 4.
King's College Publications, London, U.K.,
Modnet Preprint 14.
We exhibit a simplified exposition of Hrushovski's fusion of two strongly minimal theories
over a trivial geometry.
- Fusion over a vectorspace, with
Andreas Baudisch and Martin Ziegler,
J. Math. Logic 6 (2006), 141--162,
Modnet Preprint 39.
Given two countable strongly minimal theories with the DMP, whose common theory is the
theory of vector spaces over a fixed finite field, we show that the union of the two theories has a
strongly minimal completion.
- On fields and colors, with
Andreas Baudisch and Martin Ziegler,
Algebra and Logic 45 (2006), 92--105,
We exhibit a simplified version of the construction of a field of
Morley rank p with a predicate of rank $p-1$, extracting the main
ideas for the construction from previous papers and refining the
arguments. Moreover, an explicit axiomatization is given and ranks
- Galois cohomology of fields with a dimension, J. of Algebra 298 (2006), 34--40,
We consider fields with an abstract notion of dimension as stated by Pillay and Poizat in their
paper of 1995. We prove that for every finite extension L of K and for every finite Galois extension
L1 of L , the Brauer group of L1 over L is finite, as well as the first cohomology
group of L1 over L with coefficients in some algebraic
- Elliptic and Hyperelliptic Curves over Supersimple fields in characteristic 2, J. of pure and
applied Algebra 204 (2006), 368--379,
In this paper, we extend a previous result of A. Pillay and the author regarding existence of
rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple
fields to the even characteristic case.
- Elliptic and Hyperelliptic Curves over Supersimple fields, with
Anand Pillay, J. of London Math. Soc.
69 (2004), 1--13,
It is proved that, if F is an infinite field with characteristic different from 2, whose theory
is supersimple, and C is an elliptic or hyperelliptic curve over F with generic modulus,
then C has a generic F-rational point. The notion of genericity here is in the sense of the
supersimple field F.