5 papers · 1 filter
Confocal families of plane algebraic curves
Ragni Piene, Boris Shapiro
We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformat…
Algebraicity of exterior Cauchy transforms of algebraic ovals: a homological formulation
Ch. Hagg, B. Shapiro
Let $Ω\subset\C$ be a bounded domain whose boundary is an oval of a real algebraic curve. We study when the exterior Cauchy transform \[ \ct_Ω(z)=\frac1π\int_Ω\frac{dA(ζ)}{z-ζ} \]…
Canonical forms and moment-generating functions of plane polypols
Boris Shapiro
We study two closely related objects associated with plane domains bounded by rational algebraic arcs: canonical forms in the sense of positive geometry and normalized moment-gener…
Plane rectifiable curves: old and new
Boris Shapiro, Guillaume Tahar
In this note we recall the classical notion of an algebraically rectifiable plane curve going back to J. A. Serret, E. Laguerre and G. Humbert. We provide new criteria of algebraic…
Ribbon graphs and meromorphic functions
B. Shapiro
Let Y be a compact Riemann surface, phi:Y -> CP^1 a meromorphic function, and Gamma in Y a ribbon graph avoiding the critical points of phi. Then phi(Gamma) is an immersed graph in…