1,721,043 research outputs found
Effective Theorems for Quadratic Spaces over the Algebraic Closure of Q
No full textLet N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ QN 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 well-known theorems of Schlickewei-Schmidt and Vaaler proved respectively over Q and over a number field. We use our result to prove an effective version of Witt orthogonal decomposition for a bilinear space over Q. We also demonstrate an orthogonal version of Siegel’s lemma for a bilinear space over Q. This extends previous results of the author over a number field. All bounds on height are explicit
Small Zeros of Quadratic Forms over the Algebraic Closure of Q
Get PDFLet N \u3e= 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 \u3c= L \u3c= 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
Well-Rounded Zeta-Function of Planar Arithmetic Lattices
Get PDFWe investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at s=1 with a real pole of order 2, improving upon a result of Stefan Kühnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less than or equal to N is O(N log N) as N → ∞. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate convergence properties of a zeta-function of similarity classes of such lattices, building on the results of a paper by Glenn Henshaw, Philip Liao, Matthew Prince, Xun Sun, Samuel Whitehead, and the author
Effective Theorems for Quadratic Spaces Over Q-bar
No full textLet N \u3e=2 be an integer, F a quadratic form in N variables over Qbar, and Z contained in Qbar^N an L-dimensional subspace, 1 \u3c= L \u3c= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over Qbar 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 Qbar. If time allows, we will also discuss some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over Qbar. This extends previous results of the author over number fields. All bounds on height are explicit
Counting Lattice Points in Admissible Adelic Sets
Get PDFLecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005
Math in the Grocery Aisle: From Stacking Oranges to Constructing Error-correcting Codes
No full textThis talk was given as an invited lecture at GEMS: Gateway to Exploring Mathematical Sciences, at the Claremont Colleges in February 2009
Lattices Must be Fat - No Skinny Lattice for You!
No full textA lattice of rank k in n-dimensional Euclidean space has a shortest basis, which possesses many important properties and figures prominently in discrete optimization and theoretical computer science. In particular, it must satisfy a certain near-orthogonality condition: the angle between every pair of vectors in this basis must be between 60 and 120 degrees. This fact goes back to the work of Lagrange and Gauss. More generally, consider a collection of m \leq k vectors from a shortest basis, and let A be the solid angle that they span. How small can A be? Same as in the case m=2, there are reasons to believe that perhaps it cannot be too small, which means that lattices should have relatively fat layers, in a certain sense. While easily accessible for m=2, this question turns out to be much more difficult when m \u3e 2. I will discuss some recent results in this direction when m=3, and exhibit a connection of this question to the classical kissing number problem of Gregory and Newton. This is joint work with Sinai Robins
On Similarity Classes of Well-Rounded Sublattices of Z²
No full textA lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a non-commutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on the distribution of well-rounded lattices. We also construct a sequence of similarity classes of well-rounded sublattices of Z2, which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define. Finally, we discuss distribution of similarity classes of well-rounded sublattices of Z2 in the set of similarity classes of all well-rounded lattices in R2
On Heights of Algebraic Numbers
No full textWeil height h of an algebraic number z measures its arithmetic complexity , and h(z) is always non-negative. In fact, h(z) = 0 if and only if z is a root of unity. So suppose z is an algebraic number of degree d which is not a root of unity. How small can h(z) be? A famous conjecture of D. H. Lehmer (1932) states that h(z) cannot be arbitrarily close to 0, in fact there is (conjecturally) a gap between 0 and the smallest height value of an algebraic number of degree d, where this gap depends on d. There are many results in the direction Lehmer\u27s conjecture, although the conjecture is still open. We will discuss Lehmer\u27s conjecture, some related results, and a fascinating development of Zhang, Zagier, and others (mid-90\u27s) on height restrictions for points on certain curves
Heights and Effective Theory of Quadratic Forms over Global Fields
No full textA celebrated theorem of Cassels (1955) asserts that an integral quadratic form, which is isotropic over Q, has a non-trivial integral zero of small size (explicitly bounded), where the size is measured by a naive height function: the maximum of absolute values of the coordinates of the point in question; the bound is in terms of the height of the coefficient vector of the quadratic form. In the later years, analogues of Cassels\u27 result have been proved over other global fields: over number fields by Raghavan (1975), over rational function fields by Prestel (1987), and over over algebraic function fields by Pfister (1997). Further extensions of Cassels\u27 theorem to small-height isotropic subspaces of a quadratic space, using the contemporary theory of height functions, have been obtained by Schlickewei over Q (1985) and by Vaaler over number fields (1987). More recently, there has also been work on effective (with respect to height) decompositions of bilinear spaces, as well as further generalization of this theory to the situations with additional algebraic conditions and even over quaternion algebras. In this talk, I will give a survey of this lively area, starting from Cassels\u27 original result and up until the recent developments
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