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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 ±à¼­ÊÖ¼Ç