The Chebotarev Density Theorem and the representation of infinitely many numbers by forms

Question

Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the proofs of this result is elementary and the other, which is more oldest, is based on the Chebotarev density theorem.

My question is if there is other similar results, based on the Chebotarev density theorem, and implying the existence of forms (with degree more than $2$, more variables than $2$) representing infinitely many prime numbers (or other family of numbers).

I would know if there is any other significant results like the theorem above, where the Chebotarev density theorem must be a fundamental ingredient of their proof. I need references.

Thank you for any help.

Answer 1

The example of Heath-Brown's article is a good one. For a bit more elementary examples, you can fix a number field $K$ with $[K : \mathbb{Q}] = n$ and ring of integers $\mathcal{O}_{K}$ and pick a basis $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$. (I will assume that $n$ is odd for convenience - this guarantees the existence of a unit of norm $-1$.) Then, let $$ P(x_{1}, x_{2}, \ldots, x_{n}) = N_{K/\mathbb{Q}}\left(\sum_{i=1}^{n} x_{i} \alpha_{i}\right). $$ The polynomial $P$ will have integer coefficients and be a degree $n$ form in $n$ variables. The polynomial $P$ will represent a prime number $p$ if and only if there is a principal prime ideal in $\mathcal{O}_{K}$ with norm $p$. The existence of infinitely many such ideals follows from the Chebotarev density theorem. (It seems likely to me a modification of this construction can obtain forms corresponding to non-trivial classes in the ideal class group, but I don't see how right now.)

For example, if $K = \mathbb{Q}[\sqrt[3]{-7}]$, then we can take $$ P(x,y,z) = x^{3} + 7y^{3} + 49z^{3} - 21xyz.$$ The ring of integers in $K$ is $\mathcal{O}_{K} = \mathbb{Z}[\sqrt[3]{-7}]$ has class number $3$, and $P$ represents the primes $7$, $29$, $41$, $71$, $83$, $\ldots$ that split in the Hilbert class field of $K$. The number of such primes $\leq x$ is asymptotic to $\frac{2}{9} \frac{x}{\log(x)}$.