منابع مشابه
Small Zeros of Quadratic Forms
Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a wellknown theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bil...
متن کاملSmall Zeros of Quadratic Forms over Q
Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bi...
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 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 کاملApplications of quadratic D-forms to generalized quadratic forms
In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2012
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-2012-11310-3