Variations on methods of Lorentz and Lorentz for dimensions two and three

Abstract

This dissertation will address the problem of polynomial interpolation: finding a polynomial P(x), which goes through points pi with multiplicity mi at each point. Although polynomials are the building block for many numerical methods, such as finite elements and splines, and theorems about approximation of functions or numerical schemes almost always reduce to local interpolation by polynomials, the theory is underdeveloped. The general problem of computing the dimension of a space of polynomials satisfying certain multiplicity conditions at a set of general points can be formulated in any dimension. This problem, in its most general form, is still unsolved. The only statement known in higher dimension involves the multiplicity two case, which was solved in 1988 by J. Alexander and A. Hirschowitz. Their approach is from an algebraic geometry point of view. In this dissertation I will discuss this problem and present an alternate approach to the theorem, which I believe to be much more accessible than that given by Alexander and Hirschowitz. Throughout the paper I will use a slight variation of the methods developed by R.A. Lorentz and G.G. Lorentz, with which they have shown the dimension two case. In the last chapter I apply the methods developed thus far to toric surfaces. I start with a complete analysis of linear systems in P1 x P1, followed by a discussion of convex polygons, which correspond to certain toric compactifications of A2. In each case I describe how the method is applied and what the exceptional cases are.