Аннотация:
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 $h_{n}$.