Théorie des nombres et applications C.I.R.M., 30 novembre — 4 décembre 2009

Jacques Martinet. Hermite versus Minkowski. Let Λ be a lattice in an n-dimensional Euclidean space E, of determinant D. Define the

invariant H(Λ) (resp. M(Λ)) as the minimum of (\frac{N(e_{1})···N(e_{n})}{D})^{1/n} where e_{1}, …, e_{n} ∈ Λ constitute a basis for Λ (resp. for E) and then their supremum H_{n} and M_{n} on the set of all lattices. Upper bounds for H_{n} (resp. M_{n}) were given by Hermite (resp. by Minkowski); Minkowski’s bound is simply the Hermite constant which has been intensively studied since it was defined. In the talk, we shall prove that H_{n}/M_{n} is equal to max(1, \frac{n}{4}) for n ≤ 8, a result conjectured by Achill Schürmann, and which might well be still correct for n = 9. (For n > 9, H_{n} is strictly larger than \frac{n}{4}.)