Primality of multiply connected polyominoes
Abstract
It is known that the polyomino ideal of simple polyominoes is prime. In this paper, we focus on multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2019
- DOI:
- 10.48550/arXiv.1907.08438
- arXiv:
- arXiv:1907.08438
- Bibcode:
- 2019arXiv190708438M
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Combinatorics;
- 13A02 05E40
- E-Print:
- In this version we proved that the grid polyominoes are primes without the use of Groebner basis (see previous version). In particular, we prove that the polyomino ideal is equal to the toric ideal J_P associated to the polyomino as we defined in Section 3