## Discriminantly separable polynomials and quad-graphs    [PDF]

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. From a geometric point of view, such a classification is related to the types of pencils of conics. We construct also discrete integrable systems on quad-graphs associated with the discriminantly separable polynomials. We establish a relationship between our classification and the classification of integrable quad-graphs which has been suggested recently by Adler, Bobenko and Suris. As a fit back benefit, we get a geometric interpretation of their results in terms of pencils of conics. In the case of general position, when all four zeros of the polynomial \$P\$ are distinct, we get a connection with the Buchstaber-Novikov two-valued groups on \$\mathbb {CP}^1\$.
View original: http://arxiv.org/abs/1303.6534