We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which d-dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in d variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal product with the power series presenting the perfect coloring is zero. Using known results we obtain a sufficient condition for forced periodicity of two-dimensional perfect colorings. As corollaries of this result we get simple new proofs for known results of forced periodicity on the square and the triangular grids. Moreover, we obtain a new result concerning forced periodicity of perfect colorings in the king grid. We also consider perfect colorings of a particularly simple type: configurations that have low abelian complexity with respect to some shape, and we generalize a result that gives a sufficient condition for such configurations to be necessarily periodic. Also, some algorithmic aspects are considered.

我们研究了在某些约束下二维构型的强制周期性，并使用代数方法来研究多维符号动力学，其中d维构型和有限模式分别表示为形式幂级数和洛朗多项式，在d中变量。我们认为完美着色是一种配置，使得任何点的邻域中给定颜色的点数仅取决于某个固定相对邻域的点的颜色，并且我们表明，通过适当地选择字母表，任何完美着色都具有一个非平凡的歼灭子，即存在一个洛朗多项式，其与呈现完美着色的幂级数的形式积为零。使用已知的结果，我们获得了二维完美着色的强制周期性的充分条件。作为这个结果的推论，我们得到了关于正方形和三角形网格上强制周期性的已知结果的简单的新证明。此外，我们获得了关于王网格中完美着色的强制周期性的新结果。我们还考虑特别简单类型的完美着色：相对于某些形状具有低阿贝尔复杂度的配置，并且我们概括了一个结果，该结果给出了此类配置必然具有周期性的充分条件。此外，还考虑了一些算法方面。