Prime Number Theorems for Polynomials from Homogeneous Dynamics

We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of $n\times n$$n\times n$ matrices, the Pfaffian on the space of skew-symmetric $2n\times 2n$$2n\times 2n$ matrices, and the determinant polynomial on the space of symmetric $n\times n$$n\times n$ matrices. In particular, let $(V,F)$$(V,F)$ be any pair among the following: $({\mathrm{M}\mathrm{a}\mathrm{t}}_n,det)$$(\textrm{Mat}_{n}, \det )$, $({\mathrm{S}\mathrm{k}\mathrm{e}\mathrm{w}}_{2n},\mathrm{P}\mathrm{f}\mathrm{f})$$(\textrm{Skew}_{2n},\textrm{Pff})$, and $({\mathrm{S}\mathrm{y}\mathrm{m}}_n,det)$$(\textrm{Sym}_{n}, \det )$. We then obtain an asymptotic for

$${\pi }_{V,F}(T)=\mathrm{\#}\{v\in V:max(|v_i|)\le T,F(v)\text{ is prime}\},$$ $$\pi _{V,F}(T)= \#\{v\in V: \max (|v_{i}|)\leq T, F(v) \text{ is prime}\},$$

that matches the Bateman-Horn prediction.

The key ingredients of our proof are an asymptotic count for integral points on the level sets of $F$$F$ given by Linnik equidistribution, a geometric approximation of the box by cones, and an upper bound sieve to bound the number of prime values missed by the approximation. In the case of the determinant polynomial on symmetric matrices, we must also use the Siegel mass formula to compute the product of local densities for the main term.