Program
Wednesday 7th June
14h30-15h30: F. Amoroso
15h45-16h15: coffee break at Caffé Risorgimento (*)
16h15-16h45: L. Terracini (part I)
17h-17h45 : M. Weimann
Thursday 8th June
9h30-10h: L. Terracini (PART II)
10h15-10h45: T. Rivoal
11h-11h30: coffee break at Caffé Risorgimento (*)
11h30-12h: L. Capuano
12:30: Lunch
après-midi: collaborations
20h00: dinner
Friday 9th June
9h30-10h: M. Mula
10h15-11h15: G. Rémond
(*) Address: Caffé Risorgimento, Via Carlo Alberto 11f, 10123, Torino (just in fron of the Math Department)
TITLES AND ABSTRACTS
Francesco Amoroso (Université Caen-Normandie)
Diophantine Geometry and lacunary polynomials
Résumé : The application of arithmetic geometry to effective algebra is relatively new. We can quote for example, after the founding work of Andrzej Schinzel in the years 1950-80, the use of lower bounds for the height to the factorization of lacunary polynomials in one (Lenstra 1999) or several variables (Kaltofen - Koiran 2005 and Avendao - Krick - Sombra 2007). As an application of one of the few known instances of Zilber's conjecture, M. Filaseta, A. Granville and A. Schinzel (2008) showed that we can compute, in almost polynomial time in the the degree, the gcd of two polynomials without cyclotomic factors and with fixed integer coefficients (but with unbounded degree). More recently these results have been generalised to questions of complexity related to overdetermined systems of polynomial equations (A. - Leroux - Sombra 2015), to the search for multiple roots of lacunary polynomials (A. - Sombra - Zannier 2017) and to the factorization of lacunary polynomials over function fields (A. - Sombra 2017). Some progress on explicit results have been recently obtained by A. and Andriamandratomanana.
In this lecture we give an overview of these results, describing the underlying arithmetic geometry tools: height lower bounds, Zilber's conjecture, Zannier's toric version of Bertini's theorem.
Laura Capuano (Università di Roma tre)
On p-adic contued fractions over number fields
Résumé : It is a classical property of real continued fractions that only rational numbers have finite continued fraction expansion and, if the number is irrational, the expansion provides the best rational approximations of the number. Motivated by this property, many mathematicians tried to give a “good" definition of p-adic continued fractions, but in this context many problems arises, first of all a good definition of a floor function with the same properties of the classical one. In this talk I will give an overview on different definitions of p-adic continued fractions available in the literature and their different properties.
Marzio Mula (Università di Trento)
From the heights to the floor - Continued fractions on different algebraic structures
Résumé : Continued fractions have classically been defined over Q and number fields. However, the definition of continued fractions can actually be extended to any division algebra equipped with a metric. We explore two cases of algebraic interest: indefinite quaternion algebras over Q and infinite algebraic extensions. While the fundamental definitions and convergence properties of continued fractions readily carry over to these new settings, ensuring finite (possibly short) continued fraction expansions of global elements (CFF property) seems more challenging. Notably, the natural generalizations of Browkin's p-adic continued fractions fall short to satisfy the CFF property. For quaternion algebras, we characterize finite continued fractions expansions by means of a suitable notion of height. For infinite extensions, we can "favor" the CFF property by tweaking in various ways the definition of floor function on which continued fractions rely.
Gaël Rémond (Université Grenoble-alpes, CNRS)
Siegel fields and Northcott fields
Résumé : A Northcott field is a subfield of the algebraic numbers with no infinite subset of bounded height. A Siegel field is a subfield over which a Siegel lemma holds. I will review the definitions and the known examples then sketch the ideas of the proof of the main result relating these two notions (obtained jointly with Éric Gaudron) namely that the fields which are both Siegel and Northcott are exactly the number fields.
Tanguy Rivoal (Université Grenoble-alpes, CNRS)
On Abel's problem
Résumé : Abel's problem is the following: given an algebraic function n(x), determine when the differential equation y'(x)=n(x)y(x) admits a non-trivial algebraic solution y(x). I will present a complete solution to this problem when x*n(x) is also assumed to be a hypergeometric series with rational parameters. This is a joint work with Eric Delaygue (Lyon).
Lea Terracini (Università di Torino)
Bogomolov property for some modular Galois representations
Résumé : In 2013 P. Habegger proved the Bogomolov property for the field generated over $\mathbb{Q}$ by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions fixed by the kernel of some modular Galois representations.
Martin Weiman (Université Caen-Normandie)
About the hyperflex locus of hypersurfaces
Résumé :A point p of a projective hypersurface V of P^n is a flex point if there is a line with unexpected contact order k>= n+1 with V at p. This generalises the classical inflexion point of a curve. In collaboration with Busé, D'Andrea and Sombra, we determined the degree and an explicit equation of the flex locus of V, with a particular focus on the generic situations. Our proofs are based on the theory of multidimensional resultants. I will briefly explain our results et I will sketch the possibilities and difficulties to generalize our result to the "hyperflex" case k>=n+2 (ambitious project in collaboration with Cristina Bertone).