| There is a good inequality between the height of a point and the height of the image of the point by given morphism. When we have a rational map, then it is invalid because the functorial property fails. However, we can find a weaker inequality by introducing a new invariant of a rational map, the D-ratio. By observing the geometric definition of the degree of a morphism, we define the D-ratio of a rational map. The D-ratio of a rational map will serve an important role in height inequalities for a rational map as the degree does for a morphism. In Chapter 2, the author gives preliminaries including the resolution of indeterminacy. In Chapter 3, the author defines An -effectiveness of divisors, which is essential in defining the main concept of the dissertation, and the D-ratio, the main idea of this work. We give some applications to height inequalities in Chapter 4 and to arithmetic dynamics in Chapter 5. We have a detailed example of a regular affine automorphism in Chapter 6. Finally, the author introduces further questions and future work in Chapter 7. |
