On forms in prime variables

Abstract:

Let $F_1$, …, $F_R$ be homogeneous polynomials of degree $d\geqslant 2$ with integer coefficients in $n$ variables, and let $\mathbf {F}=(F_1,\ldots ,F_R)$. Suppose that $F_1$, …, $F_R$ is a non-singular system and $n\geqslant 4^{d+2}d^2R^5$. We prove that there are infinitely many solutions to $\mathbf {F}(\mathbf {x})=\mathbf {0}$ in prime coordinates if (i) $\mathbf {F}(\mathbf {x})=\mathbf {0}$ has a non-singular solution over the $p$-adic units $\mathbb {U}_p$ for all prime numbers $p$, and (ii) $\mathbf {F}(\mathbf {x})=\mathbf {0}$ has a non-singular solution in the open cube $(0,1)^n$.