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 Mat. Zametki, 2013, Volume 94, Issue 5, Pages 648–660 (Mi mz9274)

A Criterion for the Integral Equivalence of Two Generalized Convex Integer Polyhedra

A. V. Bykovskaya

M. V. Lomonosov Moscow State University

Abstract: We introduce the notion of integral equivalence and formulate a criterion for the equivalence of two polyhedra having certain special properties. The category of polyhedra under consideration includes Klein polyhedra, which are the convex hulls of nonzero points of the lattice $\mathbb Z^3$ that belong to some $3$-dimensional simplicial cone with vertex at the origin, and therefore the criterion enables one to improve some results related to Klein polyhedra. In particular, we suggest a simplified formulation of a geometric analog of Lagrange's theorem on continued fractions in the three-dimensional case.

Keywords: integral equivalence of polyhedra, generalized convex polyhedron, Klein polyhedron, three-dimensional continued fraction.

DOI: https://doi.org/10.4213/mzm9274

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English version:
Mathematical Notes, 2013, 94:5, 609–618

Bibliographic databases:

UDC: 511.4
Revised: 23.11.2012

Citation: A. V. Bykovskaya, “A Criterion for the Integral Equivalence of Two Generalized Convex Integer Polyhedra”, Mat. Zametki, 94:5 (2013), 648–660; Math. Notes, 94:5 (2013), 609–618

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9274
• https://doi.org/10.4213/mzm9274
• http://mi.mathnet.ru/eng/mz/v94/i5/p648

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This publication is cited in the following articles:
1. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
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