Complanart of polynomial equations

Abstract

In this paper we study polynomial maps of vector spaces and their eigenvectors and eigenvalues. The new quantity called complanart is defined. Complanarts determine complanarity of solution vectors of systems of polynomial equations. Evaluation of complanart is reduced to evaluation of resultants. As in linear case, the pattern of eigenvectors defines the phase diagram of associated differential equation. Theory of such differential equations arise naturally as extension of Lyapunov's theory of stability for solutions of differential equations. The results of this work have a number of potential applications: from solving non-linear differential equations and calculating non-linear exponents to taking non-Gaussian integrals.