Gröbner bases of balanced polyominoes

Abstract

Abstract. We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a squarefree Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced.

The coordinate ring of a simple polyomino

“…As a consequence we obtain that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain. This result covers the case of row or column convex polyominoes as well as of treelike polyominoes which are treated in [7]. We also would like to mention that there are some examples of polyominoes with holes whose coordinate rings nevertheless are not domains.…”

“…A classification of convex polyominoes whose polyomino ideal is linearly related is given in [4]. In a subsequent paper [7] of Qureshi with Shikama and the first author of this paper, balanced polyominoes were introduced. To define a balanced polyomino, one labels the vertices of a polyomino by integer numbers in a way that row and column sums are zero along intervals that belong to the polyomino.…”

“…Since the lattice ideal of a saturated lattice is always a prime ideal it follows that K[P] is a domain if P is balanced. Actually in [7] it is even shown that K[P] is a normal Cohen-Macaulay domain if P is balanced.…”

“…In [7] it is conjectured that a polyomino is balanced if and only if it is simple. A polyomino is called simple if it is hole-free.…”

Toric representations of algebras defined by certain nonsimple polyominoes

“…Let P be a polyomino and K be a field. We denote by I P , the polyomino ideal attached to P, in a suitable polynomial ring over K. It is natural to investigate the algebraic properties of I P depending on shape of P. The classes of polyominoes whose polyomino ideal is prime have been discussed in many papers, including [3,2,4,6] .…”