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Articles 1 - 16 of 16
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 …
On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky
On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky
CMC Faculty Publications and Research
Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, …
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 …
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
Finding Zeros Of Rational Quadratic Forms, John F. Shaughnessy
CMC Senior Theses
In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.
Lattice Point Counting And Height Bounds Over Number Fields And Quaternion Algebras, Lenny Fukshansky, Glenn Henshaw
Lattice Point Counting And Height Bounds Over Number Fields And Quaternion Algebras, Lenny Fukshansky, Glenn Henshaw
CMC Faculty Publications and Research
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number …
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN where N≥ 2. Let ZK be a union of varieties defined over K such that V ⊈ ZK. We prove the existence of a point of small height in V \ ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing ZK, where dependence on …
Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
Integral Orthogonal Bases Of Small Height For Real Polynomial Spaces, Lenny Fukshansky
CMC Faculty Publications and Research
Let PN(R) be the space of all real polynomials in N variables with the usual inner product < , > on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in PN(R) of degree ≤ M. We exhibit two applications of this formula. First, given a finite dimensional subspace V of PN(R) defined over Q, we prove the existence of an orthogonal basis for (V, < , >), consisting of polynomials of small height …
Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the AMS Special Session on Number Theory, October 2008.
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 …
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
CMC Faculty Publications and Research
Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.
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.
Heights And Diophantine Problems, Lenny Fukshansky
Heights And Diophantine Problems, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at Rice University, September 2004.
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.