Klein's Polyhedra for the Forth Extremal Cubic Form.

Abstract: Davenport and Swinnerton-Dyer had found the first 19 extremal thernar cubic forms gi, the meaning of which is the same as the meaning of the Markov forms for binary quadratic forms. The Klein's polyhedra for the forms g1,g2,g3were recently computed by Bruno and Parusnikov. They computed convergents of continued fractions for some matrix generalizations of the continued fractions algorithm as well. In this paper the analogious problems for the form g4are studied. Namely, the Klein polyhedra for the form g4and its conjugated oneˆg*4are computed. It appears that they are essentially different. Their periods and fundamental domains were found. The matrix algorithm's expansions of the vectors of these forms are computed as well.Note: Research direction:Mathematical problems and theory of numerical method