Klein’s polyhedral for two Davenport’s cubic forms
Abstract:
In 1895 F. Klein had suggested a plane geometric interpretation of continued fractions and their following space generalization. Three planes going through zero divide the three-dimensional space info octants ΟΣ. In each of ΟΣ we take the convex hull KΣ of all integer points except zero. The integer points of the boundary of the polyhedron KΣ (especially vertexes) must give the best rational approximations to the mentioned planes. We suggest a method for computation of the boundary gKΣ of the KΣ by means of its dual polyhedron [1].
In 1938-1943 H.Davenport had found two ternary cubic forms g₁(Χ) and g₂(Χ) which are product of three real homogenous linear forms with the unit determinant . In integer Χ≠0 the minimal values of |g₁(Χ)| and |g₂(Χ)| are minimal of possible and equal to 1/7 and 1/9 correspondingly. Here the Klein’s polyhedral for these forms were computed. They are two- periodical. We have found their automorphysms and fundamental domains.
Publication language:russian, pages:32
Research direction:
Mathematical problems and theory of numerical methods
