# Properties of high rank subvarieties of affine spaces

## TL;DR

This paper investigates high rank polynomial families in high-dimensional affine spaces over finite fields, establishing algebraic and geometric properties, and applying these to problems like the Stillman conjecture and polynomial extension.

## Contribution

It introduces new properties of high rank polynomial systems and applies them to solve longstanding problems in algebraic geometry over finite fields.

## Key findings

- Proved properties of high rank polynomial systems.
- Derived an effective version of the Stillman conjecture.
- Showed extension of weakly polynomial functions to polynomials.

## Abstract

We use tools of additive combinatorics for the study of subvarieties defined by {\it 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 [5]. 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.