Heights and quadratic forms: Cassels’ theorem and its generalizations
نویسنده
چکیده
In this survey paper, we discuss the classical Cassels’ theorem on existence of small-height zeros of quadratic forms over Q and its many extensions, to different fields and rings, as well as to more general situations, such as existence of totally isotropic small-height subspaces. We also discuss related recent results on effective structural theorems for quadratic spaces, as well as Cassels’-type theorems for small-height zeros of quadratic forms with additional conditions. We conclude with a selection of open problems.
منابع مشابه
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,...
متن کاملSmall Zeros of Quadratic Forms Outside a Union of Varieties
Let F be a quadratic form in N ≥ 2 variables defined on a vector space V ⊆ KN over a global field K, and Z ⊆ KN be a finite union of varieties defined by families of homogeneous polynomials over K. We show that if V \ Z contains a nontrivial zero of F , then there exists a linearly independent collection of small-height zeros of F in V \ Z, where the height bound does not depend on the height o...
متن کاملTotally Isotropic Subspaces of Small Height in Quadratic Spaces
Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F ) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effecti...
متن کاملA new characterization for Meir-Keeler condensing operators and its applications
Darbo's fixed point theorem and its generalizations play a crucial role in the existence of solutions in integral equations. Meir-Keeler condensing operators is a generalization of Darbo's fixed point theorem and most of other generalizations are a special case of this result. In recent years, some authors applied these generalizations to solve several special integral equations and some of the...
متن کاملSmall Zeros of Quadratic Forms with Linear Conditions
where H here stands for height of x and F , respectively. This generalizes a well known result of Cassels [2] about the existence of small zeros of quadratic forms with rational coefficients to the existence of small zeros of quadratic polynomials with rational coefficients. We generalize Masser’s result in the following way. Let K be a number field of degree d over Q. Let the coefficients fij ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012