On the exponential transform of lemniscates

It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general are not rational functions, of any form. Several examples are given to illustrate the general picture. The main tool used is that of polynomial and meromorphic resultants.Comment: 19 pages, to appear in the Julius Borcea Memorial Volume, (eds. Petter Branden, Mikael Passare and Mihai Putinar), Trends in Mathematics, Birkhauser Verla

Vladimir I. Arnold: collected works : hydrodynamics, bifurcation theory, algebraic geometry : 1965-1972

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his ""Collected Works"" focuses on hydrodynamics, bifurcation theory, and algebraic geometry

Resolving Singularities of Plane Analytic Branches With One Toric Morphism

Let (C; 0) be an irreducible germ of complex plane curve. Let \Gamma ae N be the semigroup associated to it and C \Gamma ae C g+1 the corresponding monomial curve, where g is the number of Puiseux exponents of (C; 0). We show, using the specialization of (C;0) to (C \Gamma ; 0), that the same toric morphisms Z (\Sigma) ! C g+1 which induce an embedded resolution of singularities of (C \Gamma ; 0) also resolve the singularities of (C; 0) ae (C g+1 ; 0), the embedding being defined by elements of the analytic algebra OC;0 whose valuations generate the semigroup \Gamma