Hermite interpolation 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 $\bf Q$-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 $n\geqslant 2$ 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 $N_n(H)$.

number theory

Audience: researchers in the topic

( slides )

Harvard number theory seminar