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 $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, 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}

Upper bounds for the number of solutions to quartic Thue equations

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve (x,y) that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki [2] by giving special treatments to forms with respect to their signature

Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering

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 $k$ be a number field. We consider norm form equations associated to a full $O_k$-module contained in a finite extension field $l$. 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 Thue equations fail the integral Hasse principle

For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ 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