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Ai(x American Mathematical Society An(x analytic analytic function arbitrary Axiom coefficients congruence conjugate conjugate system consider constant contains convergent coordinates corresponding cubic cubic curve curvature curve defined definition degree denote derivatives determined differential divisible divisor domain double points equal equation exists expression factor finite norm finite number fixed follows formula function genus geodesic given Hence homogeneous homogeneous coordinates ideal identity infinite inflexion points intersection interval limit point linear Math matrix modulo multiplication nilpotent algebra obtain pair permutation plane polynomial positive integer power series prime prime ideals proof proved quadratic quadric rational rational function real points region replace respectively result root of unity roots satisfied singular point solution square substitutions Suppose surface tangent Theorem tion transformation triangle Type uniformly convergent unit circle unity values vanish variables zero