A d-dimensional configuration \(c:\mathbb {Z}^d\longrightarrow A\) is a coloring of the d-dimensional infinite grid by elements of a finite alphabet \(A\subseteq \mathbb {Z}\). The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a \((d-1)\)-dimensional linear subspace \(S\subseteq \mathbb {R}^d\) is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all \((d-1)\)-dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of \(\mathbb {Z}^d\) by translations of a single tile.

d维配置\(c:\mathbb {Z}^d\longrightarrow A\)是通过有限字母表\(A\subseteq \mathbb {Z}\)的元素对d维无限网格进行着色。如果c 的有限多个平移的非平凡线性组合是零配置，则配置c 具有零配置。将c写为d变量形式幂级数，歼灭子可以方便地表示为d变量洛朗多项式f，其与c的形式积是零功率级数。更一般地，如果形式乘积是强周期性配置，我们将多项式f称为c的周期化器。一组配置的公共消灭器（周期化器）称为该集合的消灭器（分别为周期化器）。特别是，我们考虑d维子移位的消灭器和周期器，即通过不允许某些局部模式定义的配置集。我们证明，如果子移位具有一个其支持正好包含S的一个元素的周期器，则\((d-1)\)维线性子空间\(S\subseteq \mathbb {R}^d\)对于子移位是可扩展的。由于已知子移位是有限的，如果所有\((d-1)\)维子空间是可扩展的，我们在周期器上获得了一个简单的必要条件，它保证了子移位的有限性，或者等效地，配置的强周期性。我们通过单个图块的翻译来提供\(\mathbb {Z}^d\)平铺的示例。