An Infinite Series of Perfect Quadratic Forms and Big Delaunay Simplices in~$\mathbb Z^n$

R. M. Erdahl ; K. A. Rybnikov

Informatics and Automation, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 170-178

Résumé

G. Voronoi (1908–09) introduced two important reduction methods for positive quadratic forms, the reduction with perfect forms and the reduction with $L$-type domains. A form is perfect if it can be reconstructed from all representations of its arithmetic minimum. Two forms have the same $L$-type if the Delaunay tilings of their lattices are affinely equivalent. Delaunay (1937–38) asked about possible relative volumes of lattice Delaunay simplices. We construct an infinite series of Delaunay simplices of relative volume $n-3$, the best known up to now. This series gives rise to an infinite series of perfect forms with remarkable properties (e.g. $\tau_{5}\sim D_{5}\sim\phi _{2}^{5}$, $\tau _{6}\sim E_{6}^{\ast }$, and $\tau _{7}\sim \varphi _{15}^{7}$); for all $n$, the domain of $\tau _{n}$ is adjacent to the domain of $D_{n}$, the $2$nd perfect form. The perfect form $\tau _{n}$ is a direct $n$-dimensional generalization of the Korkine and Zolotareff $3$rd perfect form $\phi _{2}^{5}$ in five variables. We prove that $\tau _{n}$ is equivalent to the Anzin (1991) form $h_{n}$.