Properties of High Rank Subvarieties of Affine Spaces

We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional 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 X ⊂ A n of high rank any weakly polynomial function on a set X ( k ) ⊂ k n extends to a polynomial.