Chapter Twenty - Polynomial Discriminants Part II, Tensor Diagrams: January–February 2001

Abstract

This chapter describes the usage of diagrams to compute discriminants of polynomials and to solve a related problem, for example, line-curve tangency. Two-dimensional homogeneous geometry uses three-element vectors, 3×3 matrices, 3×3×3 tensors, and so forth, to represent various objects. For example, a homogeneous point P is a three-element row vector, and a line L is a three-element column vector. The point lies on the line if the dot product P·L is zero. Einstein index notation differentiates between two types of indices for vector/matrix elements: the point-like ones (that is expected to be known as contravariant and will be written as superscripts) and the line-like ones (would be known as covariant and also written as subscripts). Thus an element of a point-vector is Pi and an element of a line-vector is Li.