Akhtari, ShabnamDethier, Christophe2020-12-082020-12-082020-12-08https://hdl.handle.net/1794/25875In this manuscript we study the family of diagonalizable forms, a special family of integral binary forms. We begin with a summary of definitions and known results relevant to binary forms, diagonalizable forms, Thue equations, and reduction theory. The Thue--Siegel method is applied to derive an upper bound on the number of solutions to Thue's equation $F(x,y) = 1$, where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in the argument to handle forms whose discriminant is small in absolute value. These results are applied to bound the number of integral points on a certain family of elliptic curves. A proof is given for an alternative classification of diagonalizable forms using the Hessian determinant. Algebraic restrictions are given on the coefficients of a diagonalizable form and divisibility conditions are given on its discriminant. A reduction theory for the family of diagonalizable forms is given. This theory is used to computationally verify that $F(x,y) = 1$, where $F$ is a quintic diagonalizable form with small discriminant, has few solutions.en-USAll Rights Reserved.Diophantine ApproximationDiophantine EquationsInvariant TheoryNumber TheoryReduction TheoryThue EquationsElectronic Thesis or Dissertation