Shabnam Akhtari

Representation of unity by binary forms

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree

Cubic Thue inequalities with positive discriminant

n \geq 3

UPPER BOUNDS FOR THE NUMBER OF SOLUTIONS TO QUARTIC THUE EQUATIONS

, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at most 11n-2 solutions in integers x and y. We will also establish some sharper bounds when more restrictions are assumed. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential role in studying the geometry of our Diophantine equations.

The Diophantine equation aX 4 – bY 2 = 1

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}.

Lower Bounds for Heights in Relative Galois Extensions

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve

On the height of solutions to norm form equations

\phi(x, y)

Minkowski’s theorem on independent conjugate units

that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of authors earlier work with Okazaki by giving special treatments to forms with respect to their signature.

Cubic Thue equations

Abstract As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX 4 – bY 2 = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp.

Quartic Thue Equations

The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number

The method of Thue-Siegel for binary quartic forms

\alpha