Number Theory Seminar

Speaker: Jean-Marc Couveignes (Universite de Bordeaux)

Title: Representing and counting number fields

Abstract:

There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a Q-basis, the traces of a large enough family of elements, etc. From any  way of specifying  a number field one can hope to deduce  a bound on the number N_n(H) of number fields of given degree n  and  discriminant bounded by H. Experimental data suggest that the number of isomorphism classes of number fields of degree n and discriminant bounded by H is equivalent to c(n)H when n ≤2 is fixed and H tends to infinity. Such an estimate has been proved for n=3 by Davenport and  Heilbronn and for n=4, 5 by Bhargava. For an arbitrary n Schmidt  proved a bound of the form c(n)H^(n+2)/4 using Minkowski's theorem. Ellenberg et Venkatesh have proved that the exponent of H in N_n(H) is less than sub-exponential in log(n). I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for N_n(H).

The talk will take place on Zoom. If you would like to receive the Zoom link and are not part of our current NT Seminar mailing list, please contact the organizer.