Complanart of polynomial equations

TL;DR

This paper introduces the complanart, a new mathematical quantity for polynomial maps, which helps analyze the geometry of solutions and extends stability theory to nonlinear differential equations.

Contribution

It defines the complanart and demonstrates its role in understanding solution vectors and stability patterns in polynomial systems, extending classical linear concepts.

Findings

Complanart determines the complanarity of solution vectors.

Evaluation of complanart reduces to calculating resultants.

Pattern of eigenvectors influences the phase diagram of related differential 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.