Birch's theorem with shifts

TL;DR

This paper extends Birch's theorem to shifted rational forms, proving density and asymptotic solution counts for shifted forms under certain conditions, with implications for understanding the distribution of values of polynomial systems.

Contribution

It generalizes Birch's theorem to include shifts by an irrational number, establishing density and asymptotic formulas for solutions of shifted polynomial systems.

Findings

Values of shifted forms are dense in rd2

Asymptotic count of solutions in 2-balls is established

Conditions on forms ensure non-degeneracy and density

Abstract

Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{rank}(\nabla f_k)_{k=1}^R < R$. Assume that $\mathbf{f} = \mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \cdot \nabla f_k$ are linearly independent. Let $\boldsymbol{\tau} \in \mathbb{R}^R$, let $\mu$ be an irrational real number, and let $\eta$ be a positive real number. We consider the values taken by $\mathbf{f}(x_1 + \mu, ..., x_n + \mu)$ for integers $x_1, ..., x_n$. We show that these values are dense in $\mathbb{R}^R$, and prove an asymptotic formula for the number of integer solutions $\mathbf{x} \in [-P,P]^n$ to the system of inequalities $|f_k(x_1 + \mu, ..., x_n + \mu) - \tau_k| < \eta$ ($1 \le k\le R$).