Abstract
In this paper, we establish three results on small-height zeros of quadratic polynomials over \(\overline{\mathbb Q}\). For a single quadratic form in \(N \ge 2\) variables on a subspace of \(\overline{\mathbb Q}^N\), we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of k quadratic forms on an L-dimensional subspace of \(\overline{\mathbb Q}^N\), \(N \ge L \ge \frac{k(k+1)}{2}+1\), we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and m linear polynomials in \(N \ge m+4\) variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels’ theorem and its various generalizations and contributes to the literature of so-called “absolute” Diophantine results with respect to height. All bounds on height are explicit.
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Acknowledgements
The author thanks Wai Kiu Chan and Glenn R. Henshaw for many discussions and their helpful comments on the subject of this paper, and acknowledges the wonderful hospitality and support of the Erwin Schrödinger Institute for Mathematical Physics in Vienna, Austria, where a part of this work was done. The author is also grateful to the referee for the helpful remarks and suggestions. The author was partially supported by the Simons Foundation Grant #279155 and NSA Grant #130907.
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Fukshansky, L. Height bounds on zeros of quadratic forms over \(\overline{\mathbb Q}\) . Mathematical Sciences 2, 19 (2015). https://doi.org/10.1186/s40687-015-0038-5
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DOI: https://doi.org/10.1186/s40687-015-0038-5