The Classical Artin Approximation Theorems
نویسنده
چکیده
The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q : formal, analytic, algebraic, of systems of equations of the same quality Q, assuming the existence of power series solutions of a weaker quality Q’ < Q : approximated, formal. The results are frequently used in commutative algebra and algebraic geometry. We present a systematic argument which proves, with minor modifications, all theorems simultaneously. More involved results as e.g. Popescu’s nested approximation theorem for algebraic equations or statements about the Artin function will only be mentioned but not proven. We complement the article with a brief account on the theory of algebraic power series, two applications of approximation to singularities, and a differential-geometric interpretation of Artin’s proof. 1. What is the problem? 1 2. The various approximation theorems 3 3. The analytic and the algebraic case 9 4. The parametrized and the strong case 13 5. Weierstrass division theorem 18 6. Algebraic power series 20 7. Formal and analytic relations 24 8. Two applications of approximation 26 9. The geometry behind Artin’s proof 28
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