Abstract:
We study homogeneous polynomial maps of vector spaces $z_i\to A_i^{i_1i_2\dots i_s}z_{i_1}z_{i_2}\cdots z_{i_s}$ 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.
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