Discriminantly separable polynomials and quad-equations

TL;DR

This paper classifies degree-two discriminantly separable polynomials in three variables, linking their structure to integrable quad-equations and the classification of conic pencils, advancing understanding of algebraic and geometric integrability.

Contribution

It provides a comprehensive classification of discriminantly separable polynomials and connects this classification to integrable quad-equations and conic pencil structures.

Findings

Classification of discriminantly separable polynomials achieved

Established link between polynomial classification and integrable quad-equations

Connected polynomial zeros structure to conic pencil classification

Abstract

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler, Bobenko, and Suris.