Birch's theorem with shifts

Abstract

Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > σ+ R(R+1)(d-1)2^{d-1}$ variables, where $σ$ 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 $\boldsymbolτ \in \mathbb{R}^R$, let $μ$ be an irrational real number, and let $η$ be a positive real number. We consider the values taken by $\mathbf{f}(x_1 + μ, ..., x_n + μ)$ 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 $|f_k(x_1 + μ, ..., x_n + μ) - τ_k| < η$ ($1 \le k\le R$).