Hardy-Littlewood circle method for non-diagonal quadratic forms In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x_1, ..., x_s) = k$ for $(x_1, ..., x_s) \in \mathbb{Z}^s$, where $F$ is some indefinite integral quadratic form, and $k\neq 0$ is a fixed integer.
An old paper of Sarnak et. al. (A Proof of Siegel's Weight Formula, see here or here, for a PDF) mentions that solving such a problem can be done with the Hardy-Littlewood circle method if $s\geq 5$ (the paper mentions this in section 2, just before formula 2.2). They mention the $s=4$ case is harder, and provide a reference to a paper; I tracked down the paper they refer to, although again it seems to only cover the diagonal case. I am not very intereseted in this case; I just want to see how to solve it for $s\geq 5$ for non-diagonal forms. I found some lecture notes of a course given by Sarnack covering the case of diagonal forms for $s\geq 5,$ but I do not see how to generalize. Is there some easy way to go from knowing this result for diagonal forms to non-diagonal ones, or some other source which covers solutions to indefinite forms? The proof given by Sarnack in those notes seems to use in a fundamental way the fact that the form is diagonal, so that a certain factorization can work.
 A: The "best" way to deal with quadratic forms using the circle method is via Heath-Brown's delta symbol method.
You can read about this in detail in the paper:
Heath-Brown - A New Form of the Circle Method, and its application to Quadratic Forms
Theorem 4 in particular gives an asymptotic formula for the problem you mention. Note that Heath-Brown is also able to obtain results for 4 variables.
A: One reason to restrict to diagonal quadratic forms is that this is almost no loss of generality: Quadratic forms can always be diagonalized over $\mathbb{Q}$ by a linear change of variables.
There are several issues, but they can be resolved:

*

*Since we are interested in integer solutions, as pointed out by Dan, we need to be careful with denominators. However, the diagonalization introduces at most finitely many congruence conditions on our new variables.
This is not a big problem since your exponential sum will then factor as a product of terms like $\sum_{\vert x\vert \le P, x \equiv a (q)} e(\alpha x^2)$ and for the purposes of the circle method these can be treated just as the classical sum without the congruence condition.


*If you want a precise asymptotic for the integer solutions inside some growing box (rather than just upper and lower bounds of the correct order of magnitude), you might need to be careful because your box gets distorted by the change of variables. However, this can be resolved by an easy enveloping argument.
So all of this is fine, as long as we fix our quadratic form. If we want to obtain results that are uniform over a family of quadratic forms, for instance with an explicit dependence on the size of the coefficients of our form, this approach can become problematic and is probably not the ideal one.
