## Annals of Mathematics |

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abelian group algebraic analytic analytic functions arbitrary arithmetic mean boundary circle coefficients collinear collineation complex quantities conic conjugate consider constant contains converges coordinates corresponding cubic curvature cusp cyclic group defined denote derivatives determined differential equation divisible essential singularities exists expression finite norm following theorem formula geodesic curvature given greatest common divisor Green's function group of order Hence identically infinite number integer integral curve intersection interval invariant plane invariant three-chain inverse isogonals Jacobian linear chain linearly independent locus Math matrix monodromie group multiple necessary and sufficient obtained odd prime orthogonal osculating parameter pencil planar chain positive integer prime factor prime number proof prove quadratic quotient group rational region relation respect roots satisfy singular solution subgroup substitution sufficient condition surface Sylow subgroup symmetric tangent transformation values vanish variables zero