Birch's theorem with shifts

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 $\mathbf{f}=0$ has a nonsingular real solution, and that the forms $(1,...,1)\cdot \nabla f_k$ are linearly independent. Let $\mathbf{\tau }\in {\mathbb{R}}^R$, 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 $|f_k(x_1+\mu ,...,x_n+\mu )-{\tau }_k|<\eta$ ($1\le k\le R$).

Slight changes from version 1 based on comments from the referee