The Smallest Solution of an Isotropic Quadratic Form

An isotropic quadratic form f(x1,...,xn) = ∑ ni=1 ∑ nj=1 fijxixj defined on a Z- lattice has a smallest solution, where the size of the solution is measured using the infinity norm (∥ ∥∞), the l1 norm (∥ ∥1), or the Euclidean norm (∥ ∥2). Much work has been done to find the least upper bound and greatest lower bound on the smallest solution, beginning with Cassels in the mid-1950’s. Defining F := (f11,...,f1n,f21,...,f2n,...,fn1,...,fnn), upper bound results have the form ∥x∥i ≤ C∥F∥iθ , with i ∈ {1, 2, ∞} and C a constant depending only on n. Aside from Cassels and Davenport, authors have concentrated more on finding the smallest exponent θ and less on C. Since Cassels’ publication, others have generalized his result and answered related questions. In particular, Schlickewei and Schmidt considered cases in which a quadratic form vanishes on a space of dimension greater than 1 and whether or not there is a bound on the product of the norms of multiple linearly independent nontrivial solutions. Schulze-Pillot explored similar questions, in addition to considering bounds on the determinant of the matrix whose columns are linearly independent nontrivial solutions of a quadratic form. Another goal has been to determine the best possible bound in each of these cases. In particular, Kneser demonstrated Cassels’ bound is best possible and Schlickewei and Schmidt have shown their results, along with several of Schulze-Pillot’s, are best possible. This dissertation gives a detailed, comprehensive exposition of these results. In particular, many of the results of Schulze-Pillot and Schlickewei and Schmidt are re- examined, leading to greater insight into the details of the proofs. The main methods used to solve this problem come from quadratic form theory, lattice theory, and the geometry of numbers