We study multidimensional configurations (infinite words) and subshifts of low pattern complexity using tools of algebraic geometry. We express the configuration as a multivariate formal power series over integers and investigate the setup when there is a non-trivial annihilating polynomial: a non-zero polynomial whose formal product with the power series is zero. Such annihilator exists, for example, if the number of distinct patterns of some finite shape D in the configuration is at most the size $|D|$ of the shape. This is our low pattern complexity assumption. We prove that the configuration must be a sum of periodic configurations over integers, possibly with unbounded values. As a specific application of the method we obtain an asymptotic version of the well-known Nivat's conjecture: we prove that any two-dimensional, non-periodic configuration can satisfy the low pattern complexity assumption with respect to only finitely many distinct rectangular shapes D.

我们使用代数几何工具研究多维配置（无限词）和低模式复杂度的子移位。我们将配置表示为整数的多元形式幂级数，并在存在非平凡的an灭多项式时进行研究：一个非零多项式，幂级数的形式积为零。例如，如果配置中某些有限形状D的不同图案的数量最多为$|d|$的形状。这是我们低模式复杂度的假设。我们证明该配置必须是整数（可能具有无限制值）上的周期性配置的总和。作为该方法的一种特定应用，我们获得了著名的Nivat猜想的渐近形式：我们证明了任何二维，非周期性的配置都只能满足有限的多个复杂矩形D的低图案复杂性假设。