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On extreme forms in dimension 8

Cordian Riener (2006)

Journal de Théorie des Nombres de Bordeaux

A theorem of Voronoi asserts that a lattice is extreme if and only if it is perfect and eutactic. Very recently the classification of the perfect forms in dimension $8$ has been completed [5]. There are 10916 perfect lattices. Using methods of linear programming, we are able to identify those that are additionally eutactic. In lower dimensions almost all perfect lattices are also eutactic (for example $30$ out of the $33$ in dimension $7$ ). This is no longer the case in dimension $8$ : up to similarity, there...

On Gauss's proof of Seeber's Theorem.

G. Rousseau (1992)

Aequationes mathematicae

On perfect forms in six variables.

В.С. Владимиров (1958)

Matematiceskij sbornik

On the computation of Hermite-Humbert constants for real quadratic number fields

Michael E. Pohst, Marcus Wagner (2005)

Journal de Théorie des Nombres de Bordeaux

We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields $ℚ (\sqrt{13})$ and $ℚ (\sqrt{17})$ . Finally we compute the Hermite-Humbert constant for the number field $ℚ (\sqrt{13})$ .

On the S-Euclidean minimum of an ideal class

Kevin J. McGown (2015)

Acta Arithmetica

We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. In particular, we resolve a conjecture of Lenstra except when the S-units have rank one. The proof is self-contained but uses ideas from...

One-class genera of positive quaternary quadratic forms

G. Watson (1974)