29 research outputs found
Integral points on a certain family of elliptic curves
The Thue-Siegel method is used to obtain an upper bound for the number of
primitive integral solutions to a family of quartic Thue's inequalities. This
will provide an upper bound for the number of integer points on a family of
elliptic curves with j-invariant equal to 1728
The Method Of Thue-Siegel For Binary Quartic Forms
We will use Thue-Siegel method, based on Pad\'e approximation via
hypergeometric functions, to give upper bounds for the number of integral
solutions to the equation as well as the inequalities , for a certain family of irreducible quartic binary forms.Comment: A version of this paper is to appear in Acta. Arit
Cubic Thue inequalities with positive discriminant
We will give an explicit upper bound for the number of solutions to cubic
inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer
coefficients and positive discriminant D. Our upper bound is independent of h,
provided that h is smaller than D^{1/4}
A positive proportion of Thue equations fail the integral Hasse principle
For any nonzero , we prove that a positive proportion of
integral binary cubic forms do locally everywhere represent but do not
globally represent ; that is, a positive proportion of cubic Thue equations
fail the integral Hasse principle. Here, we order all classes of
such integral binary cubic forms by their absolute discriminants. We prove
the same result for Thue equations of any fixed degree ,
provided that these integral binary -ic forms are ordered by the maximum
of the absolute values of their coefficients.Comment: Previously cited as "A positive proportion of locally soluble Thue
equations are globally insoluble", Two typos are fixed and small mathematical
error in Section 4 is correcte
Heights, Regulators and Schinzel's determinant inequality
We prove inequalities that compare the size of an S-regulator with a product
of heights of multiplicatively independent S-units. Our upper bound for the
S-regulator follows from a general upper bound for the determinant of a real
matrix proved by Schinzel. The lower bound for the S-regulator follows from
Minkowski's theorem on successive minima and a volume formula proved by Meyer
and Pajor. We establish similar upper bounds for the relative regulator of an
extension of number fields.Comment: accepted for Publication in Acta Arit
On the height of solutions to norm form equations
Let be a number field. We consider norm form equations associated to a
full -module contained in a finite extension field . It is known that
the set of solutions is naturally a union of disjoint equivalence classes of
solutions. We prove that each nonempty equivalence class of solutions contains
a representative with Weil height bounded by an expression that depends on
parameters defining the norm form equation
A positive proportion of locally soluble quartic Thue equations are globally insoluble
For any fixed nonzero integer , we show that a positive proportion of
integral binary quartic forms do locally everywhere represent , but do
not globally represent .
We order classes of integral binary quartic forms by the two generators of
their ring of -invariants, classically denoted by
and .Comment: 17 pages, to appear in Mathematical Proceedings of the Cambridge
Philosophical Societ