## On algebraically integrable Birkhoff and angular billiards. (arXiv:1706.04030v1 [math.DS])

We present a solution of the algebraic version of Birkhoff Conjecture on
integrable billiards. Namely we show that every polynomially integrable real
bounded convex planar billiard with smooth boundary is an ellipse. We also
extend this result to the case of piecewise-smooth and not necessarily convex
polynomially integrable billiards: we show that the boundary is a union of
confocal conical arcs and straight-line segments lying in some special lines
defined by the foci. The proof, which is obtained by Mikhail Bialy, Andrey
Mironov and the author, is split into two parts. The first part is the paper by
Bialy and Mironov, where they prove the following theorems: 1) the polar
duality transforms a polynomially integrable planar billiard to a rationally
integrable angular billiard; 2) the singularities and inflection points of each
irreducible component of the complexified curve polar-dual to the billiard
boundary lie in the two complex isotropic lines through the origin; 3) the
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