, Let b be a branch satisfying condition (ii-a) of Theorem 4.1. Then its base point C is a regular point of the conic I, and b is tangent to I. We treat the two following cases separately. Case 1): I is a union of two lines. Then b has local relative projective symmetry property of type A-w, by Proposition 4.16, Subcase 3a). Hence, it is quadratic, by Theorem 4.17. Case 2): I is a regular conic, Hence, it is quadratic, vol.17
, Rationally integrable I-angular billiards. Proof of Theorem 1, p.25
CP 2 be a conic (regular or a pair of distinct lines), and let ? ? CP 2 be an irreducible algebraic curve different from a line and from I and generating a rationally integrable I-angular billiard ,
, All the singular and inflection points (if any) of the curve ? lie in I
, Namely, for every C 2 -smooth arc ? ? ?? with non-zero geodesic curvature the statement of Theorem 6.1 is proved there for each non-linear irreducible component ? of the Zariski closure of the ?-dual curve ? * . But the proofs given in [10, 11] remain valid in the general context of Theorem 6.1. Each local branch of the curve ? at a base point in ? ? I that satisfies the conditions of some of the statements (i), (ii-a), or (ii-b) of Theorem 4.1 also satisfies the corresponding statement
Let ?(M ) be its non-trivial homogeneous polynomial integral of even degree 2n: M = [r, v], and ?([r, v]) is not a function of the squared norm ||v|| 2 =< Av, v > in the metric of the surface ?, One has ?(M ) ? c < AM, M > n , since < AM, M >=< Av ,
Integrable billiards, Mosc. Univ. Mech. Bull, vol.45, issue.6, pp.13-17, 1990. ,
On integrable systems with elastic reflections (in Russian), Mosc. Univ. Mech. Bull, vol.45, issue.5, pp.14-16, 1990. ,
Golfer's dilemma, Am. J. Phys, vol.74, issue.6, pp.497-501, 2006. ,
An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math, issue.2, pp.527-558, 2016. ,
Caustics and evolutes for convex planar domains, J. Diff. Geometry, vol.28, pp.345-357, 1988. ,
Seules les quadriques admettent des caustiques, Bull. Soc. Math. France, vol.123, pp.107-116, 1995. ,
Convex billiards and a theorem by E, Hopf. Math. Z, vol.214, issue.1, pp.147-154, 1993. ,
Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane, Discrete Contin. Dyn. Syst, vol.33, issue.9, pp.3903-3913, 2013. ,
On totally integrable magnetic billiards on constant curvature surface, Electron. Res. Announc. Math. Sci, vol.19, pp.112-119, 2012. ,
Angular billiard and algebraic Birkhoff conjecture, Adv. in Math, vol.313, pp.102-126, 2017. ,
Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, J. Geom. Phys, vol.115, pp.150-156, 2017. ,
On fourth-degree polynomial integrals of the Birkhoff billiard, Proc. Steklov Inst. Math, vol.295, issue.1, pp.27-32, 2016. ,
Algebraic non-integrability of magnetic billiards, J. Phys. A, vol.49, issue.45, p.pp, 2016. ,
A survey on polynomial in momenta integrals for billiard problems, Issue 2131, vol.336, 2018. ,
First integrals of systems with gyroscopic forces. (Russian) Vestnik Moskov, Univ. Ser. I Mat. Mekh, vol.113, issue.6, pp.75-82, 1984. ,
, Integrable Birkhoff billiards. Mosc. Univ. Mech. Bull, vol.45, issue.2, pp.10-13, 1990.
Integrable billiards on surfaces of constant curvature, Math. Notes, vol.51, issue.1-2, pp.117-123, 1992. ,
Plane algebraic curves, 1986. ,
On Birkoff 's [Birkhoff 's] conjecture about convex billiards, Proceedings of the 2nd Catalan Days on Applied Mathematics, pp.85-94, 1995. ,
Integrable billiards and quadrics, Russian Math. Surveys, vol.65, issue.2, pp.319-379, 2010. ,
Bicentennial of the great Poncelet theorem (1813-2013): current advances, Bull. Amer. Math. Soc. (N.S.), vol.51, issue.3, pp.373-445, 2014. ,
Pseudo-integrable billiards and arithmetic dynamics, J. Mod. Dyn, vol.8, issue.1, pp.109-132, 2014. ,
Periods of pseudo-integrable billiards, Arnold Math. J, vol.1, issue.1, pp.69-73, 2015. ,
Pseudo-integrable billiards and double reflection nets, Russian Math. Surveys, vol.70, issue.1, pp.1-31, 2015. ,
On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J, vol.14, issue.2, pp.239-289, 2014. ,
On two-dimensional polynomially integrable billiards on surfaces of constant curvature, Doklady Mathematics, vol.98, issue.1, pp.382-385, 2018. ,
URL : https://hal.archives-ouvertes.fr/ensl-01964938
On polynomially integrable planar outer billiards and curves with symmetry property, Math. Annalen, vol.372, pp.1481-1501, 2018. ,
URL : https://hal.archives-ouvertes.fr/ensl-01413589
Introduction to singularities and deformations, 2007. ,
Principles of algebraic geometry, vol.1, 1978. ,
Arithmetic genera and effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto, Sect, vol.30, pp.177-195, 1956. ,
On local Birkhoff Conjecture for convex billiards, Ann. of Math, vol.188, issue.1, pp.315-380, 2018. ,
Symmetries and topology of dynamical systems with two degrees of freedom, Russian Acad. Sci. Sb. Math, vol.80, issue.1, pp.105-124, 1995. ,
A genetic introduction to the dynamics of systems with impacts, Translated from Russian by J.R.Schulenberger. Translations of Mathematical Monographs, vol.89, 1991. ,
The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija, vol.7, pp.185-214, 1973. ,
Entropy of billiard maps and a dynamical version of the Birkhoff conjecture, J. Geom. Phys, vol.124, pp.413-420, 2018. ,
Singular points of complex hypersurfaces, 1968. ,
The billiard ball problem on a table with a convex boundary -an illustrative dynamical problem, Ann. of Math, issue.2, pp.446-470, 1950. ,
Integrable curvilinear billiards, Phys. Lett. A, vol.115, issue.1-2, pp.25-28, 1986. ,
On invariants of singular points of algebraic curves, Math. Notes, vol.34, issue.5-6, pp.962-963, 1983. ,
Geometry and billiards. Student Mathematical Library, Mathematics Advanced Study Semesters, vol.30, 2005. ,
On algebraically integrable outer billiards, Pacific J. of Math, vol.235, issue.1, pp.101-104, 2008. ,
Billiard map and rigid rotation, Phys. D, pp.31-34, 2013. ,
On a Conjugacy Problem in Billiard Dynamics, Proc. Steklov Inst. Math, vol.289, issue.1, pp.291-299, 2015. ,
A locally integrable multi-dimensional billiard system, Discrete Contin. Dyn. Syst, vol.37, issue.10, pp.5271-5284, 2017. ,
Integrable systems with discrete time, and difference operators, Funct. Anal. Appl, vol.22, issue.2, pp.83-93, 1988. ,
Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys, vol.7, pp.81-107, 1990. ,
Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom, vol.40, issue.1, pp.155-164, 1994. ,