Alexandre Eremenko

Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry

Suppose that 2d - 2 tangent lines to the rational normal curve z → (1: z …: z d ) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.

Pole Placement by Static Output Feedback for Generic Linear Systems

We consider linear systems with m inputs, p outputs, and McMillan degree n such that n=mp. If both m and p are even, we show that there is a nonempty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that, for each system in U, there exists an open set of pole configurations, symmetric with respect to the real line, which cannot be assigned by any real static output feedback.

Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers θ 1 , θ 2 , θ 3 for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles 2πθ 1 , 2πθ 2 , 2πθ 3 . The same condition is necessary and sufficient for the triple πθ 1 , πθ 2 , πθ 3 to be interior angles of a spherical triangular membrane.

Meromorphic solutions of higher order Briot–Bouquet differential equations

For differential equations P ( y ( k ) , y )=0, where P is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.

Rational functions and real Schubert calculus

We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.

ON THE AREA DISTORTION BY QUASICONFORMAL MAPPINGS

We give the sharp constants in the area distortion inequality for quasiconformal mappings in the plane.

High Energy Eigenfunctions of One-Dimensional Schrödinger Operators with Polynomial Potentials

For a class of one-dimensional Schrödinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit distribution in the complex plane as the eigenvalues tend to infinity. This limit distribution depends only on the degree of the polynomial potential and on the boundary conditions.

On the zeros of meromorphic functions of the formf(z)=Σ k=1 ∞ a k/z−z k

We study the zero distribution of meromorphic functions of the formf(z)=Σk=1∞ak/z−zk whereak>0. Noting thatf is the complex conjugate of the gradient of a logarithmic potential, our results have application in the study of the equilibrium points of such a potential.Furthermore, answering a question of Hayman, we also show that the derivative of a meromorphic function of order at most one, minimal type has infinitely many zeros.

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as