Lattice Embeddings of Planar Point Sets
Autor: | Jesse Milzman, Derek A. Smith, Dara Zirlin, Dantong Zhu, Michael Knopf |
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Rok vydání: | 2016 |
Předmět: |
Discrete mathematics
Coprime integers 010102 general mathematics Lattice (group) Principal ideal domain 0102 computer and information sciences 01 natural sciences Ring of integers Theoretical Computer Science Combinatorics Computational Theory and Mathematics Integer 010201 computation theory & mathematics Discrete Mathematics and Combinatorics Geometry and Topology Ideal (ring theory) 0101 mathematics Complex plane Mathematics Heronian triangle |
Zdroj: | Discrete & Computational Geometry. 56:693-710 |
ISSN: | 1432-0444 0179-5376 |
Popis: | Let $$\mathcal {M}$$M be a finite non-collinear set of points in the Euclidean plane, with the squared distance between each pair of points integral. Considering the points as lying in the complex plane, there is at most one positive square-free integer D, called the "characteristic" of $$\mathcal {M}$$M, such that a congruent copy of $$\mathcal {M}$$M embeds in $$\mathbb {Q}(\sqrt{-D})$$Q(-D). We generalize the work of Yiu and Fricke on embedding point sets in $$\mathbb {Z}^2$$Z2 by providing conditions that characterize when $$\mathcal {M}$$M embeds in the lattice corresponding to $$\mathcal {O}_{-D}$$O-D, the ring of integers in $$\mathbb {Q}(\sqrt{-D})$$Q(-D). In particular, we show that if the square of every ideal in $$\mathcal {O}_{-D}$$O-D is principal and the distance between at least one pair of points in $$\mathcal {M}$$M is integral, then $$\mathcal {M}$$M embeds in $$\mathcal {O}_{-D}$$O-D. Moreover, if $$\mathcal {M}$$M is primitive, so that the squared distances between pairs of points are relatively prime, and $$\mathcal {O}_{-D}$$O-D is a principal ideal domain, then $$\mathcal {M}$$M embeds in $$\mathcal {O}_{-D}$$O-D. |
Databáze: | OpenAIRE |
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