Height Bounds on Zeros of Quadratic Forms Over Q-bar

In this paper we establish three results on small-height zeros of quadratic polynomials over Q. For a single quadratic form in N ≥ 2 variables on a subspace of Q , 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 Q , N ≥ L ≥ 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 ≥ 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.