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?

  • $\begingroup$ The obvious generalization is to $n$-ary quadratic forms. They still satisfy the Hasse principle, as shown by Hasse-Minkowski. In general, the circle method can deal with degree $d$ forms as long as $n$ is large with respect to $d$ (see for example the seminal work of Birch). Another generalization is to norm-form equations; here the observation is that binary quadratic forms are norm forms for quadratic field extensions of the rationals. $\endgroup$ – Stanley Yao Xiao Aug 31 '15 at 22:59
  • $\begingroup$ Pretty cool however I am not familiar with results you are talking about? Could you pleaase post a full address of details? $\endgroup$ – user76479 Sep 1 '15 at 0:58
  • $\begingroup$ You can think of a lot of ideas. Better more clearly define the issue. For example what kind of numbers required. Although this form can always lead to the Pell equation. $\endgroup$ – individ Sep 1 '15 at 6:03
  • $\begingroup$ @StanleyYaoXiao Could you elaborate your idea? $\endgroup$ – user76479 Sep 5 '15 at 22:22
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    $\begingroup$ @StanleyYaoXiao Isn't that only for rational numbers, not integral? $\endgroup$ – Will Sawin Sep 7 '15 at 3:03

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