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Let $q$ be a indefinite quadratic form in $n$ variables with integrals coefficients.

Is there a generic (i.e. algorithmic) way to determine wether the form $q$ has non trivial integral zeros , that is if there exists some $x\in \mathbb{Z}^n - \{0\}, q(z) = 0$? And if there are some, how to find them?

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  • $\begingroup$ If $n \geq 5,$ there always are nontrivial integral zeros. When $n=4$ there usually are, the exceptions are related to quaternions. For $n=3$ it is encyclopediaofmath.org/index.php/Legendre_theorem After that, really depends on your background. For $n=2$ it is just square discriminant. Recommend Cassels, Rational Quadratic Forms. $\endgroup$ – Will Jagy Sep 1 '18 at 19:05
  • $\begingroup$ Cassels, pages 41-44 for specifics of the Hilbert Norm Residue Symbol, then pages 55-59 for isotropy. He does a separate discussion of Legendre's theorem, pages 80-82 $\endgroup$ – Will Jagy Sep 1 '18 at 19:18
  • $\begingroup$ In addition, an indefinite rational quadratic form represents zero iff it represents zero $p$-adically for all primes $p$. In effect there will be only finitely many $p$ that need consideration, and for each $p$ it will be a finite computation. As Will points out for $n\ge5$ this is all automatic. (It helps to work over $\Bbb Q$ and to diagonalise the form). $\endgroup$ – Lord Shark the Unknown Sep 1 '18 at 19:30
  • $\begingroup$ I see, thanks for the reference. My questions was for $n$ large so there always be at least one non trivial zero. But now how can you find such a zero? $\endgroup$ – TomTom Sep 1 '18 at 20:21

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