Boris Shapiro: Results and conjectures on positive algebraic Cauchy transforms

An irreducible algebraic function F in C (or, equivalently, an irreducible polynomial in 2 variables) is called a positive Cauchy transform if there exists a positive measure with compact support whose Cauchy transform coincides with a branch of F almost everywhere in C. Such measures can be considered a natural 'motherbody measures' in this context. We present interesting examples of such polynomials and a large number of conjectures based on the previous studies of (generalized) polynomial eigenfunctions to linear differential operators with polynomial coefficients. Joint with Rikard Boegvad (Stockholm university).