_{1}

We show that a rectangle can be signed tiled by ribbon
*L n*-ominoes, n odd, if and only if it has a side divisible by
*n*. A consequence of our technique, based on the exhibition of an explicit Gr
öbner basis, is that any k-inflated copy of the skewed
*L n*-omino has a signed tiling by skewed
*L n*-ominoes. We also discuss regular tilings by ribbon
*L n*-ominoes,
*n* odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.

In this article, we study tiling problems for regions in a square lattice by certain symmetries of an L-shaped polyomino. Polyominoes were introduced by Golomb in [

Tilings by

replicating tiles introduced in [

The results in [

We investigate in this paper tiling properties of

A useful tool in the study of signed tilings is a Gröbner basis associated to the polynomial ideal generated by the tiling set. If the coordinates of the lower left corner of a cell are (α, β), one associates to the cell the monomial

associates to any bounded tile placed in the square lattice a Laurent polynomial with all coefficients 1. The polynomial associated to a tile P is denoted by

Our main result is the following:

Theorem 1. A rectangle can be signed tiled by

Theorem 1 is proved in Section 4 using a Gröbner basis for the tiling set computed in Section 3.

For completeness, we briefly discuss regular tilings by

Theorem 1 gives for regular tilings by

Theorem 2. If

If one of the sides of the rectangle is divisible by n, we recall first the following result of Herman Chau, mentioned in [

Theorem 3. A rectangle with both sides odd cannot be tiled by

If one of the sides of the rectangle is even, one has the following result.

Theorem 4. Let

1) If one side is divisible by n and the other side is of even length, then the rectangle can be tiled by

2) If the side divisible by n is of length at least

Proof. 1) The rectangle can be tiled by

2) We use the tiling shown in

A consequence of the technique used in the proof of Theorem 1 is:

Proposition 5. If

Proposition 5 is proved in Section 5.

As any

Proposition 6. Let

1) If

2) If

Proposition 6 is proved in Section 6. Proposition 6 leaves open the question of replication of the skewed L n-omino if k is odd and not divisible by n. Some cases can be solved by using Pak’s higher invariants

For completeness, we also consider the tile set

Theorem 7. If

Theorem 7 is proved in Section 7.

Barnes developed in [

Theorem 8. If complex or rational weights are allowed to replace the integral weights, a rectangle can be signed tiled by

Signed tilings by

We make a final comment about the paper. While the methods that we use are well known, and algorithmic when applied to a particular tiling problem, here we apply them to solve simultaneously an infinite collection of tiling problems.

An introduction to signed tilings can be found in the paper of Conway and Lagarias [

Let

For

Definition 1. Let

It is clear that if

Definition 2. A D-Gröbner basis is a finite set G in

Proposition 9. Let G be a finite set of

1) G is a Gröbner basis.

2) Every

Note that if R is only a (PID), the normal form of the division of f by G is not unique. We introduce now the notions of S-polynomial and G-polynomial.

Definition 3. Let

Remark. If

Theorem 10. Let G be a finite set of

Assume now that R is a Euclidean domain with unique remainders (see page 463 [

Definition 4. Let

Proposition 11. E-reduction extends D-reduction, i.e., every D-reduction step in an E-reduction step.

Theorem 12. Let R be an Euclidean domain with unique remainders, and assume G is a finite set of

1)

2) E-reduction modulo G has unique normal forms.

The following result connects signed tilings and Gröbner bases. See [

Theorem 13. A polyomino P admits a signed tiling by translates of prototiles

We write

We show in the rest of this section that a Gröbner basis for the ideal generated in

It is convenient to look at the elements of the basis geometrically, as signed tiles, see

position in the square lattice to another. See

Proposition 14.

Proof. The geometric proofs appear in

Proposition 15.

Proof. We first show that

Using (3), the RHS of Equation (5) becomes:

After we obtain

A step by step geometric proof of formula (5) for

Proposition 16.

Proof. This follows from Propositions 14, 15.

Proposition 17. One has the following D-reductions

Consequently,

Proof. The leading monomial of

We show now that all above reductions are D-reductions by looking at the elimination of the terms of highest degree in the S-polynomials.

The terms of highest degrees in

are (in this order)

The terms

which does not contains terms of higher degree then

The remaining terms

which also does not contain terms of higher degree then

The term of highest degrees in

is

which does not contain terms of higher degree then

The term of highest degrees in

is

As all higher coefficients are equal to 1, we do not need to consider the G-polyno- mials.

Consider a

is divisible by the polynomial:

If

Assume that

If r is odd, one has the following sequence of remainders, each remainder written in a separate pair of parentheses:

If

If r is even, one has the following sequence of remainders, each remainder written in a separate pair of parentheses:

If

Consider a k-inflated copy of the L n-omino. Using the presence of

1) We employ a ribbon tiling invariant introduced by Pak [

for any tile in

2) Let

where we start with

In the second case, the sequence of r encodings of the ribbon tiles starts as above, but now the subsequence of zeroes reaches the left side. Then we have a jump of

So in both cases the

It is enough to generate the tile consisting of a single cell. We show the proof for

In this section we give a proof of Theorem 8 following a method developed by Barnes. The reader of this section should be familiar with [

infinite collection of tiling sets

Let

where

Separate x from

Eliminating the denominators gives:

which can be factored as:

It is clear that all roots of the polynomial above, and of the corresponding polynomial in the variable x, are roots of unity of order

We show now that I is a radical ideal. For this we use an algorithm of Seidenberg which can be applied to find the radical ideal of a zero dimensional algebraic variety over an algebraically closed field. See Lemma 92 in [

If

Proposition 18. The polynomials

Proof. It is enough to generate

We can apply now the main result in Lemma 3.8, [

which clearly evaluates to zero in all points of V if and only if one of

The fact that Theorem 8 implies Theorem 1 follows the idea of Theorem 4.2 in [

We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Grӧbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes.

V. Nitica was partially supported by Simons Foundation Grant 208729.

Nitica, V. (2016) Signed Tilings by Ribbon L n-Ominoes, n Odd, via Grӧbner Bases. Open Journal of Dis- crete Mathematics, 6, 297-313. http://dx.doi.org/10.4236/ojdm.2016.64025