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Canadian Number Theory Association VIII Meeting
June 20-25, 2004
The Fields Institute
Toronto, ON, Canada

Organizers
John Friedlander (Toronto) and Cam Stewart (Waterloo)

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Comparison of several generalizations of the continued fraction
by
Vladimir Parusnikov
Keldysh Institute of Applied Mathematics of RAS, Miusskaja sq. 4, Moscow 125047, Russia

Here are results of the comparison of generalizations of the continued fraction, given by the Klein polyhedra, by the Euler, Jacobi, Poincare, Brun and other matrix algorithms, by the Voronoi relative minima and by the Bruno polyhedra.

Davenport (1943) and Svinnerton-Dyer (1971) found first 19 extremal ternary cubic forms hi being products of three real homogenous linear forms. They have the similar sense as the well known Markov forms for binary quadratic forms. In [1-3] (1994-2000) for the first 7 extremal forms h1 - h7 and for four nonextremal forms we computed the Klein (1895)-Skubenko (1971)-Arnold (1993) polyhedra. Except that, the multiple root vectors of these forms were expanded by means of the matrix algorithms proposed by Euler (1775), Jacobi (1868), Poincare (1884), Brun (1919), Bruno (1994) and Parusnikov (1994). It appeared that each algorithm has a vector with a nonperiodic expansion and that all algorithms except Bruno's one go under the surface the Klein polyhedra. The Poincare algorithm is the worst from that view-point.

In [4] (2003) for a product of three linear homogeneous forms h(X)=l1(X)l2(X)l3(X), Bruno proposed to consider the surface DM of the convex hull of points (|l1(X)|, |l2(X)|, |l3(X)|) with integers X =/= 0. For the same eleven cubic forms, we computed the polyhedral surface DM, which are simpler than the Klein polyhedra, and found all Voronoi (1896) relative minima, some of which are inside the Bruno polyhedra.



References

1. A.D. Bruno and V.I. Parusnikov // Mathem. Notes 56: 3-4 (1994), 994-1006; 61: 3 (1997) 278-286; 67: 1 (2000) 87-102.

2. V.I. Parusnikov. Preprints of the Keldysh Inst. Appl. Math. N 137 (1996), N 93 (1997), N 36 and 69 (1998), N 69 and 79 (1999) (in Russian).

3. V.I. Parusnikov. Klein polyhedra for complete decomposable forms // Number theory. Dyophantine, Computational and Algebraic Aspects. Editors: K. Györy, A. Pethö and V.T. Sós. De Gruyter. Berlin, New York. 1998, p. 453-463.

4. A.D. Bruno and V.I. Parusnikov. Preprints of the Keldysh Inst. Appl. Math. N 86 and 93 (2003) (in Russian).

Date received: April 12, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # calz-43.