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

Georges 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 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 an infintie series of perfect forms with remarkable properties: e.g. τ5 ∼ D5, τ6 ∼ E∗ 6 , τ7 ∼ φ15; for all n the domain of τn is adjacent to the domain of Dn, the 2-nd perfect form . Perfect form τn is a direct n-dimensional generalization of Korkine and Zolotareff’s 3-rd perfect form φ2 in 5 variables. We prove that τn is equivalent to Anzin’s (1991) form hn.