Klein polyhedra for the fourth extremal cubic form
Let \(L=A\mathbb{Z}^3\) be a three-dimensional lattice with determinant \(d(L)=|\det A|=|\det(a_{ik}) |>0\) and consider the homogeneous minimum \[ \mu(L)= \inf \Biggl\{|\prod^3_{i=1} \biggl(\sum^3_{k=1} a_{ik}u_k\biggr)|: (u_1, u_2,u_3)\in \mathbb{Z}^3\setminus \{0\} \Biggr\}. \] Davenport and Swinnerton-Dyer found the absolute maximum and a sequence of isolated maxima in the spectrum of values of \(\mu(L)\). They are attained by certain lattices which are generated by totally real algebraic number fields. Continuing earlier work concerning the first three maxima and the associated extremal ternary cubic forms [\textit{A. D. Bryuno (Bruno)} and \textit{V. I. Parusnikov}, Mat. Zametki 56, No. 4, 9-27 (1994); translated in Math. Notes 56, No. 4, 994-1007 (1994; Zbl 0844.11047), \textit{N. I. Parusnikov}, Klein polyhedra for the third extremal cubic form, Preprint No. 242-137, M. V. Keldysh Institute for Appl. Math., Russian Academy of Sciences, Moscow 1996 (details not available for the reviewer); see also \textit{E. I. Korkina}, Proc. Steklov Inst. Math. 209, 124-144 (1995); translation from Tr. Mat. Inst. Steklova 209, 143-166 (1995; Zbl 0883.11034)], the author studies the Klein polyhedra associated with the fourth form and its conjugate form.
lattices generated by real number fields
scientific article; zbMATH DE number 1517725