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Abstract. 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.

Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps Z 2 → A where A is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let Pc(m, n) denote the number of distinct m × n block patterns occurring in a configuration c. Configurations satisfying Pc(m, n) ≤ mn for some m, n ∈ N are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic. Recently, Kari and the author showed that low complexity configurations can be decomposed into a sum of periodic configurations. In this paper we show that if there are at most two components, Nivat's conjecture holds. As a corollary we obtain an alternative proof of a result of Cyr and Kra: If there exist m, n ∈ N such that Pc(m, n) ≤ mn/2, then c is periodic. The technique used in this paper combines the algebraic approach of Kari and the author with balanced sets of Cyr and Kra.

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