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We consider multivariable polynomials over a fixed number field, linear in some of the variables. For a system of such polynomials satisfying certain technical conditions we prove the existence of search bounds for simultaneous zeros with respect to height. For a single such polynomial, we prove the existence of search bounds with respect to height for zeros lying outside of a prescribed algebraic set. We also obtain search bounds in the case of homogeneous multilinear polynomials, which are related to a so-called "sparse" version of Siegel's lemma. Among the tools we develop are height inequalities that are of some independent interest. [ABSTRACT FROM AUTHOR]

Given an isotropic quadratic form over a number field which assumes a value t , we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value t. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value t. [ABSTRACT FROM AUTHOR]

Abstract: Text: Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of , . Let be a union of varieties defined over K such that . We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing , where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) . As a part of our argument, we provide a basic extension of the function field version of Siegel''s lemma (Thunder, 1995) to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R-o6lr8s0Go. [Copyright &y& Elsevier]

Abstract: Text: A 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 . 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 , 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 , 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 in the set of similarity classes of all well-rounded lattices in . Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=q3LJV4lv0PA. [Copyright &y& Elsevier]

Abstract: A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of , as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of . Our results extend automatically to well-rounded sublattices of any lattice , where A is an element of the real orthogonal group . [Copyright &y& Elsevier]

Abstract: Let K be a number field, and let W be a subspace of , . Let be subspaces of of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and . Our main tool is a counting estimate we prove for the number of points of a subspace of inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]