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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Totally Isotropic Subspaces Of Small Height In Quadratic Spaces, Wai Kiu Chan, Lenny Fukshansky, Glenn Henshaw
Totally Isotropic Subspaces Of Small Height In Quadratic Spaces, Wai Kiu Chan, Lenny Fukshansky, Glenn Henshaw
CMC Faculty Publications and Research
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 effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are …
Height Bounds On Zeros Of Quadratic Forms Over Q-Bar, Lenny Fukshansky
Height Bounds On Zeros Of Quadratic Forms Over Q-Bar, Lenny Fukshansky
CMC Faculty Publications and Research
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 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 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 …
Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw
Small Zeros Of Quadratic Forms Outside A Union Of Varieties, Wai Kiu Chan, Lenny Fukshansky, Glenn R. Henshaw
CMC Faculty Publications and Research
Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace of the quadratic space such that is not …
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky
CMC Faculty Publications and Research
Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (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) over bar 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 bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.
Quadratic Forms And Height Functions, Lenny Fukshansky
Quadratic Forms And Height Functions, Lenny Fukshansky
CMC Faculty Publications and Research
The effective study of quadratic forms originated with a paper of Cassels in 1955, in which he proved that if an integral quadratic form is isotropic, then it has non-trivial zeros of bounded height. Here height stands for a certain measure of arithmetic complexity, which we will make precise. This theorem has since been generalized and extended in a number of different ways. We will discuss some of such generalizations for quadratic spaces over a fixed number field as well as over the field of algebraic numbers. Specifically, let K be either a number field or its algebraic closure, and …
Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky
Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005.
Small Zeros Of Quadratic Forms With Linear Conditions, Lenny Fukshansky
Small Zeros Of Quadratic Forms With Linear Conditions, Lenny Fukshansky
CMC Faculty Publications and Research
Given a quadratic form and M linear forms in N + 1 variables with coefficients in a number field K, suppose that there exists a point in KN+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.