On forms in prime variables

Abstract

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