Introduction to Non-Linear Algebra

TL;DR

This paper introduces non-linear algebra, extending linear algebra concepts to non-linear equations using discriminants and resultants, and explores their applications in Mandelbrot-set theory and phase transition models.

Contribution

It provides an accessible introduction to non-linear algebra, emphasizing resultants of non-linear maps and their connection to Mandelbrot-set theory and phase transitions.

Findings

Effective study of resultants using analytical and computational methods

Extension of eigenvalue concepts to non-linear maps

Initial steps linking non-linear algebra to Mandelbrot-set theory

Abstract

Concise introduction to a relatively new subject of non-linear algebra: literal extension of text-book linear algebra to the case of non-linear equations and maps. This powerful science is based on the notions of discriminant (hyperdeterminant) and resultant, which today can be effectively studied both analytically and by modern computer facilities. The paper is mostly focused on resultants of non-linear maps. First steps are described in direction of Mandelbrot-set theory, which is direct extension of the eigenvalue problem from linear algebra, and is related by renormalization group ideas to the theory of phase transitions and dualities.