We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional \({\mathbb {F}}_q\)-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry , such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of Bik et al. (Polynomials and tensors of bounded strength, arXiv:1805.01816). We also show that for k-varieties \({\mathbb {X}}\subset {\mathbb {A}}^n\) of high rank any weakly polynomial function on a set \({\mathbb {X}}(k)\subset k^n\) extends to a polynomial.

我们使用加法组合工具来研究由高维\（{{mathbb {F}} _ q \）-向量空间中的多项式的高级秩族定义的子变量。在本文的第一部分，我们证明多项式的高阶系统的许多性质。第二，我们利用这些性质来推导代数几何的结果，例如在代数封闭域上的有效Stillman猜想，在有限域上的变种的Nullstellensatz类似物，以及Bik等人最近的结果的加强。（极限强度的多项式和张量，arXiv：1805.01816）。我们还表明，对于k个高阶\（{\ mathbb {X}} \ subset {\ mathbb {A}} ^ n \），集合上的任何弱多项式函数\（{\ mathbb {X}}（k）\ subset k ^ n \）扩展为多项式。