Toric representations of algebras defined by certain nonsimple polyominoes

Shikama, Akihiro

doi:10.1216/jca-2018-10-2-265

Cited by 16 publications

(17 citation statements)

References 9 publications

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“…. , I s is a zig-zag walk in P. Following [4] and [20], we briefly recall some notations and a technical Lemma. A binomial f = f + − f − in a binomial ideal J in a polynomial ring over a field is called redundant if it can be expressed as a linear combination of binomials in J of lower degree.…”

Section: Figure 10mentioning

confidence: 99%

“…Nowadays the study is applied to multiply connected polyominoes, which are polyominoes with one or more holes. In [11] and [20], the authors have proved that the polyominoes obtained by removing a convex polyomino from a rectangle in N 2 are prime. In [13] the authors have introduced a very useful tool to provide a characterization of prime polyominoes.…”

Section: Introductionmentioning

confidence: 99%

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Primality of weakly connected collections of cells and weakly closed path polyominoes

Cisto,

Navarra,

Utano

2021

Preprint

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In this paper we study the primality of a new class of polyominoes, called weakly closed paths. Firstly, we prove that the inner 2-minors ideal associated to a weakly connected and simple collection of cells is the toric ideal of a bipartite and weakly chordal graph. We classify all weakly closed paths which do not contain zig-zag walks, we give suitable toric representations of the associated polyomino ideal and finally we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the polyomino ideal attached to a weakly closed path, supporting an existing conjecture.

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Section: Figure 10mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Primality of weakly connected collections of cells and weakly closed path polyominoes

Cisto,

Navarra,

Utano

2021

Preprint

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“…In [6] and [13], the authors prove that simple polyominoes, namely without holes, are prime. In [8] and [14], a family of prime polyominoes obtained by removing a convex polyomino by a given rectangle was showed. In a more recent paper [10], it is demonstrated that if the polyomino  is prime, then it should have no zig-zag walks, and it is conjectured that this is also a sufficient condition for the primality of .…”

Section: Introductionmentioning

confidence: 99%

Primality of polyomino ideals by quadratic Gröbner basis

Mascia

Rinaldo

Romeo

2022

Mathematische Nachrichten

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In this work, we provide a necessary and sufficient condition on a polyomino ideal for having the set of inner 2-minors as graded reverse lexicographic Gröbner basis, due to combinatorial properties of the polyomino itself. Moreover, we prove that when the latter holds the polyomino ideal coincides with the lattice ideal associated to the polyomino, that is the polyomino ideal is prime. As an application, we describe two new infinite families of prime polyominoes.

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“…In general, giving a complete characterization of the primality of multiply connected polyomino ideals seems to be not so easy. A family of prime polyominoes obtained by removing a convex polyomino by a given rectangle has been showed in [8] and [11].…”

Section: Introductionmentioning

confidence: 99%

“…It is known that a prime polyomino ideal is a toric ideal. In Section 3, we present a toric ideal associated to a polyomino, generalizing Shikama's definition in [11]. This toric ideal contains the polyomino ideal (see Proposition 3.1).…”

Section: Introductionmentioning

confidence: 99%

Primality of multiply connected polyominoes

Mascia,

Rinaldo,

Romeo

2019

Preprint

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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.

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Toric representations of algebras defined by certain nonsimple polyominoes

Abstract: In this paper we give a toric representation of the associated ring of a polyomino which is obtained by removing a convex polyomino from its ambient rectangle.2010 Mathematics Subject Classification. 13C05, 05E40.

Cited by 16 publications

References 9 publications

Primality of weakly connected collections of cells and weakly closed path polyominoes

Primality of weakly connected collections of cells and weakly closed path polyominoes

Primality of polyomino ideals by quadratic Gröbner basis

Primality of multiply connected polyominoes

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