0 Introduction
1 Algebraic Geometry Approach
1.1 Lagrange interpolation
1.2 Uniform Bivariate Hermite Interpolation
1.3 Dimension Problem
1.4 What is known
2 Approximation Theory Approach
2.1 Introduction and Basic Notation
2.2 Shifts and Algorithm
2.3 Triangles and Hermite Problems
3 The Alexander-Hirschowitz Theorem
3.1 Basic Definitions and the Theorem
3.2 Coalescence in Dimension 2
3.3 Dimension 2 Pyramids
3.4 Coalescence in Higher Dimension
3.5 Dimension 3 Pyramid
4 Toric Surfaces
4.1 Linear Systems of Curves in P1 X P1
4.2 Linear Systems of Curves in General Toric Surfaces
5 Future Work
5.1 Alexander-Hirschowitz Theorem and Dimension N
5.2 General Toric Surfaces
编辑手记
