Abstract In M Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions Motivated by this result and a similar result of P oski he con jectured that this remains true when we replace the ring of convergent power series by a more general ring This paper presents the state of the art on this problem aimed at non experts In particular we put a slant on the Artin Approximation Problem with con straints

ARTIN APPROXIMATION GUILLAUME ROND Abstract. In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P?oski, he con- jectured that this remains true when we replace the ring of convergent power series by a more general ring. This paper presents the state of the art on this problem, aimed at non-experts. In particular we put a slant on the Artin Approximation Problem with con- straints. Contents 1. Introduction 1 2. Artin Approximation 13 2.1. The analytic case 13 2.2. Artin Approximation and Weierstrass Division Theorem 22 2.3. Néron's desingularization and Popescu's Theorem 24 3. Strong Artin Approximation 27 3.1. Greenberg's Theorem: the case of a discrete valuation ring 27 3.2. Strong Artin Approximation Theorem: the general case 31 3.3. Ultraproducts and proofs of Strong Approximation type results 33 3.4. Effectivity of the behaviour of Artin functions: some examples 34 4. Examples of Applications 39 5. Approximation with constraints 41 5.1. Examples 42 5.2. Nested Approximation in the algebraic case 44 5.3. Nested Approximation in the analytic case 48 5.4. Other examples of approximation with constraints 53 Appendix A. Weierstrass Preparation Theorem 56 Appendix B. Regular morphisms and excellent rings 56 Appendix C. Étale morphisms and Henselian rings 58 References 61 1.

• relation then


• can also


• theorem yields


• relation between


• series over


• power series


• local ring


• any formal


• there exists