Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of integers $n\in\Bbb N$ by $$ax^2+bxy+cy^2$$

One simple theorem states that if discriminant is coprime to $n$ with $\Big(\frac{b^2-4ac}n\Big)=1$, there is at least one primitive form with same discriminant that represents $n$.

What is the state of art generalizing similar facts known in binary quadratic forms to general $m$-variate higher homogeneous forms?

What is a good reference for these facts?