Forms in prime variables and differing degrees

Abstract

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

35 pages