Height bounds on zeros of quadratic forms over \mathbb Q

2015/08/04 by Fukshansky, Lenny
#11E12 #11E39 #11G50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1508.00830

Abstract

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

Related

• Small representations of integers by integral quadratic forms
• Totally isotropic subspaces of small height in quadratic spaces
• On zeros of multilinear polynomials
• Quadratic Forms in Prime Variables with small off-diagonal ranks
• Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in\n at least five Variables