Discriminantly separable polynomials and their applications

Abstract

Discriminantly separable polynomials by definition are polynomials which discriminants are factorized as the products of the polynomials in one variable. Motivating example for introducing such polynomials is Kowalevski top, one of the most celebrated integrable system, where the so called Kowalevski’s fundamental equation appears to be such a polynomial. We introduced a whole class of systems which are based on discriminantly separable polynomials and on which the integration of the Kowalevski top may be generalized. We present also the role of the discriminantly separable polynomils in two well-known examples: the case of Kirchhoff elasticae and the Sokolov’s case of a rigid body in an ideal fluid. Also we present the classification of the discriminantly separable polynomials of degree two in each of three variable and relate this classification to the classification of pencils of conics. Another application of discriminantly separable polynomials is in integrable quad-equations introduced by Adler, Bobenko and Suris. This paper presents a short review of our results concerning these polynomials.