Concise introduction to a relatively new subject of non-linear algebra: literal extension of text-book linearalgebra 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 computerfacilities. The paper is mostly focused on resultants of non-linear maps. First steps are described in direction ofMandelbrot-set theory, which is direct extension of the eigenvalue problem from linear algebra, and is related byrenormalization group ideas to the theory of phase transitions and dualities