Forms in prime variables and differing degrees

Abstract

Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular system and $n\ge D^2 4^{D+6}R^5$. We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$. This can be viewed as a smooth local-global principle for $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ with differing degrees. It follows that, under (i) and (ii), the set of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ is Zariski dense in the set of its solutions.