  In this paper, it is shown that if F(x , y) is an irreducible binary form
with integral coefficients and degree $n \geq 3$, 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.
