The classical Artin approximation theorems

Abstract

The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q Q (i.e., formal, analytic, algebraic) of systems of equations of the same quality Q Q , assuming the existence of power series solutions of a weaker quality Q ′ > Q Q’ > Q (i.e., 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, such 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 of the theory of algebraic power series, two applications of approximation to singularities, and a differential-geometric interpretation of Artin’s proof.