Signed tilings by ribbon L n-ominoes, n even, via Groebner bases

Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\"obner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants.