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arXiv · 1603.00051

Algebraic Method in Tilings

Peter Horak·Dongryul Kim
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Abstract

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling $n$-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of $n$-space by translates of a cluster $V$ of prime size then there is a lattice tiling by $V$ as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster $V$ is lattice if $V$ generates $n$-space.

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Peter Horak, Dongryul Kim. 2016-02-29. Algebraic Method in Tilings. https://arxiv.org/abs/1603.00051

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