Klein polyhedra for three extremal cubic forms

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms $g_i$, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms $g_1-g_4$. They also computed the “convergents” for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms $g_5-g_7$ and the adjoint form $g^*_7$. Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincaré, Brun, Parusnikov, and Bryuno, are computed. The position of the “convergents of the continued fractions” with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Eulers and Poincarés algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.