Expressive curves

A nodal real algebraic curve in the affine plane is called expressive if its defining polynomial has the smallest number of critical points that is allowed by the topology of the set of real points of the curve. This notion can be viewed as a "global" counterpart of the "local" notion of a real morsification of an isolated singularity of a plane curve.

We establish a criterion for expressivity of a curve, describe several constructions that produce expressive curves, and relate their study to the combinatorics of links, quiver mutations, and planar bicolored graphs. This is joint work with E. Shustin.