Let \(F \in \mathbb{F}[x_1, \dots, x_n]\) be a homogeneous form of degree \(d \ge 2\). The author is concerned with the existence of points with prime coordinates on the variety \(V(F)=\{z \in \mathbb{C}^n : F(z) = 0\}\). It is shown that under suitable assumptions, infinitely many such points exist.
More precisely, suppose that no local obstructions exist, i.e., \(F(x) = 0\) has nonsingular solutions in \((0,1)^n\) as well as in \((\mathbb{Z}_p^\times)^n\) for all primes \(p\). Let \(V^*_F\) denote the singular locus of \(V(F)\), i.e., the affine variety \[ V_F^* = \{z \in \mathbb{C}^n : \nabla F(z) = 0\}, \] and suppose that \[ n-\dim V_F^* \ge 2^83^45^2d^3(2d-1)^2 4^d. \] Finally, let \(\Lambda^*(x) = \log p\) if \(x=p\) is a prime and \(\Lambda^*(x) = 0\) otherwise, and define the counting function \[ N_\mathcal{P}(F;X) = \sum_{x \in [0,X]^n} \Lambda^*(x_1) \cdots \Lambda^*(x_n) 1\!\!1_{V(F)}(x). \] Then, \(N_\mathcal{P}(F;X) \gg X^{n-d}.\)
The result improves dramatically upon a previous result of B. Cook and Á. Magyar [Invent. Math. 198, No. 3, 701–737 (2014; Zbl 1360.11063)], which in addition to the local conditions required that \(n - \dim V_F^* > \mathfrak{C}_d\), where \[ \mathfrak{C}_d > (t_{d-1}2^{5d}d! - 1)(d-1)^{t_{d-1}2^{5d}d!-1}, \] with \(t_j\) defined recursively by \(t_1 = 1\) and \(t_{j+1} = d^{t_j}\).