Algebraic Geometry Foundations

Abstract

In our study of selected polycons in Chap. 3 we introduced several concepts that require further investigation. Denominator polynomials were constructed from EIP of polycon boundaries. This raises several questions: When does a set of points on a curve of specified maximal order determine that curve? How do we allow for deficiency in intersection points caused by either intersection at infinity or coalescing of points that are in the general ca6e distinct? Given polynomials P and Q and curve R, what are the necessary and sufficient conditions for the existence of some b in the field of complex numbers such that $\mathrm{P} -\mathrm{ bQ} = 0$ everywhere on R? How may we generalize to polypols? Can we find basis functions for higher-degree approximation by similar techniques? Is there some unifying theory that will facilitate extension to higher-dimensional elements?