Number Theory Seminar
Speaker: Jean-Marc Couveignes (Universite de Bordeaux)
Title: Representing and counting number fields
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 Nn(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 Nn(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 Nn(H).
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