## Annals of Mathematics |

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2-sphere apply automorphism birational bivector boundary bounded bounded set chain transformation character characteristic closure algebra closure-algebraic functions coefficients completes the proof consider contains converges convex Corollary corresponding defined definition deformation degree denote determined dimensional domain elements equations equivalent Euclidean space exists field finite number follows formula free algebra functionally free given Hence Hilbert space homeomorphism ideal identity implies integral invariant irreducible isomorphic lattice Lemma manifold mapping Math matrix modular modular lattice modulo monoidal transformation multiple neighborhood normal normal subgroup obtain operator oriented polynomial positive proof of Theorem prove quadratic transformation quotient quotient ring Qv(P representation respectively result ring satisfy self-adjoint sequence simple point singular points sphere subalgebra subgroup subset Suppose surface symmetric symmetric matrix Theorem tion topological space uniformizing parameters v-fold curve values variables vector zero