An Infinite Series of Perfect Quadratic Forms and Big Delaunay Simplexes in Z^n

Abstract

George 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 can be reconstructed from all representations of its arithmetic minimum. Two forms have the same L-type if Delaunay tilings of their lattices are affinely equivalent. Delaunay (1937-38) asked about possible relative volumes of lattice Delaunay simplexes. We construct an infinite series of Delaunay simplexes of relative volume n-3, the best known as of now. This series gives rise to a new infintie series of perfect forms TF_{n} with interesting properties, e.g. TF_{5}=D_{5}, TF_{6}=E*_{6}, TF_{7}=\phi_{15}^{7}. For all n the domain of TF_{n} is adjacent to the domain of the 2-nd perfect form D_{n}. Perfect form TF_{n} is a direct n-dimensional generalization of Korkine and Zolotareff's 3-rd perfect form \phi_{2}^{5} in 5 variables. It is likely that this form is equivalent to Anzin's (1991) form h_n.