Jeffrey Lin Thunder

An asymptotic estimate for heights of algebraic subspaces

We count the number of subspaces of affine space with a given dimension defined over an algebraic number field with height less than or equal to B. We give an explicit asymptotic estimate for the number of such subspaces as B goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number, etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domains

Asymptotic estimates for the number of integer solutions to decomposable form inequalities

For homogeneous decomposable forms F ( X ) in n variables with integer coefficients, we consider the number of integer solutions

Inequalities for decomposable forms of degree +1 in variables

{\bf x}\in\mathbb{Z}^n

Volumes and diophantine inequalities associated with decomposable forms

to the inequality

Hermite's Constant for Function Fields

|F({\bf x})|\le m

Asymptotic estimates for rational points of bounded height on flag varieties

as

The number of solutions to cubic Thue inequalities

m\rightarrow\infty

On cubic Thue inequalities and a result of Mahler

. We give asymptotic estimates which improve on those given previously by the author in Ann. of Math. (2) 153 (2001), 767–804. Here our error terms display desirable behaviour as a function of the height whenever the degree of the form and the number of variables are relatively prime.

Counting subspaces of given height defined over a function field

We consider the number of integral solutions to the inequality |F(x)| < m, where F(X) E Z[X] is a decomposable form of degree n + 1 in n variables. We show that the number of such solutions is finite for all m only if the discriminant of F is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.

Algebraic integers of small discriminant

Abstract For homogeneous decomposable forms F( X ) in n variables with real coefficients, we consider the associated volume of all real solutions x ∈ R n to the inequality |F( x )|⩽1 . We relate this to the number of integral solutions z ∈ Z n to the Diophantine inequality |F( z )|⩽m in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.