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 کاملZeros of p-adic forms
A variant of Brauer’s induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1989
ISSN: 0022-314X
DOI: 10.1016/0022-314x(89)90065-6