Complanart of a system of polynomial equations

Abstract

We study homogeneous polynomial maps of vector spaces zi → A i 1 i 2 ...is i zi 1 zi 2 · · · zi s and their eigenvectors and eigenvalues. We define a new quantity called the complanart, which determines the coplanarity of the solution vectors of a system of polynomial equations. Evaluating the complanart reduces to evaluating resultants. As in the linear case, the pattern of eigenvectors/eigenvalues defines the phase diagram of the associated differential equation. Such differential equations arise naturally in attempting to extend Lyapunov's stability theory. The results in this paper can be used in a range of applications from solving nonlinear differential equations and calculating nonlinear exponents to evaluating non-Gaussian integrals.

New and old results in resultant theory

“…The resultant is defined only in the case where the number of homogeneous equations is equal to the number of variables; otherwise, we must use various more complicated quantities to describe the existence and/or degeneration of a solution. Some of these quantities are known as higher resultants or subresultants [6], and some are known as complanarts [7].…”

“…To illustrate these general formulas, we consider a specific case where n = 3 and r = 2: the system of equations has form (7). The traces are calculated using (14) and the differential operatorŝ and so on.…”