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Amador Martin-Pizarro

Ma recherche porte sur la théorie des modèles géométrique, ainsi que la théorie des modèles de corps et ses applications à la géométrie algébrique et à la théorie des nombres.

Liste de publications

  • Sur les automorphismes bornés de corps munis d'opérateurs, with Thomas Blossier and Charlotte Hardouin, to appear in Math. Research Letters, Arxiv 1505.03669.
    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 operators.
  • A Model Theoretic Study of Right-Angled Buildings, with Andreas Baudisch and Martin Ziegler, to appear in J. Eur. Math. Soc. HAL-01079813.
    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 algebraic.
  • 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, HAL-00702683.
    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, HAL-00863220.
    Elliptic curves over a supersimple eld 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 (2008), 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 (2007), 413--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 by Zilber.
  • 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, arXiv:math/0605412.
    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 are computed.
  • Galois cohomology of fields with a dimension, J. of Algebra 298 (2006), 34--40, (pdf).
    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 group G.
  • Elliptic and Hyperelliptic Curves over Supersimple fields in characteristic 2, J. of pure and applied Algebra 204 (2006), 368--379, (pdf).
    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, (pdf).
    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.