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 Mat. Zametki, 2005, Volume 77, Issue 4, Pages 566–583 (Mi mz2518)

Klein polyhedra for three extremal cubic forms

V. I. Parusnikov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/mzm2518

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English version:
Mathematical Notes, 2005, 77:4, 523–538

Bibliographic databases:

UDC: 511.36+514.172.45
Revised: 26.11.2004

Citation: V. I. Parusnikov, “Klein polyhedra for three extremal cubic forms”, Mat. Zametki, 77:4 (2005), 566–583; Math. Notes, 77:4 (2005), 523–538

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz2518
• https://doi.org/10.4213/mzm2518
• http://mi.mathnet.ru/eng/mz/v77/i4/p566

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii sb., 11:1 (2010), 68–73
2. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
3. A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65
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