Klein's Polyhedra for the Third Extremal Ternar Cubic Form

V. I. Parusnikov


Abstract: In 1938-1943 H.Davenport had found two ternary cubic forms g₁(X) and g₂(X) which are the product of three real homogenous linear forms with the unit determinant. In integer X ≠ 0 the minimal values of |g₁(X)| and |g₂(X)| are maximal of possible and equal to 1/7 and 1/9 correspondingly. In the present paper we study the form min|g₃(X)|=1/√148 for X∈ Z³\{0}. The cubic form g₃(X) is a candidate to the third place in a set, which is similar to the Lagrange-Markov spectrum for the quadratic forms. The Klein's polyhedra for g₃(X) were computed. They are two-periodical. We have found their automorphysms and fundamental domains.