Height bounds on zeros of quadratic forms over $\overline{\mathbb{Q}}$

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.

23 pages; to appear in Research in Mathematical Sciences. arXiv admin note: substantial text overlap with arXiv:1307.0564