Introduction to Non-Linear Algebra

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.