18 papers
Algebraic defect and positive mother body measures
Boris Shapiro
Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ , its irreducible equation , and a compact co…
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…
Van Vleck spectra of high-order Heun operators:\ finite-band universality and exterior asymptotics
Boris Shapiro
We study high-order analogues of the classical Heun operator of Fuchs index one, \[ \dq=\sum_{i=1}^k Q_i(z)\frac{d^i}{dz^i}, \qquad °Q_i\le i+1, \qquad °Q_k=k+1. \] For a fixed d…
Weak and strong -analogs of the Laguerre--Pólya class
D. K. Dimitrov, B. Shapiro
For we compare two natural -analogs of the Laguerre--Pólya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre--Pó…
Frenet turns
Boris Shapiro
We discuss a problem posed by A.~Agrachev asking how many times a usual circle in should be traversed to admit a deformation by curves with nowhere degenerating Frene…
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 \[ \ct_Ω(z)=\frac1π\int_Ω\frac{dA(ζ)}{z-ζ} \]…