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