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In a paper by Heath-Brown, upon having to estimate the number of solutions $(x,y)\in\mathbb Z^2\cap[-B,B]^2$ to the equation $Q(x,y)=k$ with $Q$ an integer-coefficient non-degenerate quadratic form and $k$ an integer, he simply states that by "standard facts from the theory of quadratic forms" this is $O_\varepsilon(H(Q)^\varepsilon B^\varepsilon)$, where $H(Q)$ is the maximum absolute value of $Q$'s coefficients.

I have tried searching for proofs of this online, but no source mentions any inequality of the sort. My attempts at proving it have been mostly analytical, notably by attempting to estimate the integral of the generating function $$S(\alpha,B)=\sum_{(x,y)\in\mathbb Z^2\cap[-B,B]^2}e(\alpha Q(x,y))$$ like in the circle method but nothing has come of it.

Also, his comment on "standard facts" make me think this must be an algebraic fact rather than an analytical one, but trying to limit the values of $x$ and $y$ to divisibility conditions involving $k$ haven't brought me any further.

Any help or reference would be appreciated. Thanks !

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    $\begingroup$ Good standard facts you can find in Diophantine Equations, L. J. Mordell (Academic Press) chapters $17,18$ and $19$. Also in the book Rational Quadratic Forms of J .W. S . Cassels, excelent but I don't remember the axact reference. $\endgroup$
    – Piquito
    Commented Jun 29 at 17:52
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    $\begingroup$ if the form is positive thge set of solutions is finite. If indefinite: the form has an "automorphism group" of linear mappings; the automorphisms with determinant $+1$ make a cyclic group, often there are also determinant $-1.$ Every solution comes by the group acting on the finite set of "fundamental" solutions in a region given by inequalities $\endgroup$
    – Will Jagy
    Commented Jun 29 at 20:56
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    $\begingroup$ Probably not, but it is just inequalities. Given indefinite form $ax^2 + bxy + c y^2$ with discriminant $d=b^2 - 4ac,$ which is positive but not a square, every Pell type solution to $t^2 - d s^2 = 4$ leads to an automorphism matrix $$ \left( \begin{array}{rr} \frac{t - bs}{2}& -cs \\ as & \frac{t+bs}{2} \\ \end{array} \right) $$ Meanwhile, do you have a link for the Heath-Brown article? $\endgroup$
    – Will Jagy
    Commented Jun 30 at 13:28
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    $\begingroup$ It seems what you need is Hua, Introduction to Number Theory, English translation 1982. Chapter 11, section 4, defines "primary" solutions of $a x^2 + b xy + c y^2 = k$ Then all of chapter 12 is Binary Quadratic Forms. $\endgroup$
    – Will Jagy
    Commented Jun 30 at 20:44
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    $\begingroup$ Your Heath-Brown refers to Wooley, and Wooley refers to Hua's book and a long article of his own with Vaughan. $\endgroup$
    – Will Jagy
    Commented Jun 30 at 20:51

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