The basic theory of power series

Abstract

The book is an introduction to analytic geometry and commutative algebra, from the point of view of formal and convergent power series; it stands at the postgraduate level and gives a clear presentation of all the basic results of local analytic geometry, with an equal treatment for the real and complex cases. This book should be very useful for anyone who wants to study analytic geometry, and would constitute an excellent support for a postgraduate course in this field. Contents: (I) Power series: (1) Series of real and complex numbers; (2) Power series; (3) Rückert's and Weierstrass's theorems. (II) Analytic rings and formal rings: (1) Mather's preparation theorem; (2) Noether's projection lemma; (3) Abhyankar's and Rückert's parametrization; (4) Nagata's Jacobian criteria; (5) Complexification. (III) Normalization: (1) Integral closure; (2) Normalization; (3) Multiplicity in dimension 1; (4) Newton-Puiseux's theorem. (IV) Nullstellensätze: (1) Zero sets and zero ideals; (2) Rückert's complex Nullstellensatz; (3) The homomorphism theorem; (4) Risler's real Nullstellensatz; (5) Hilbert's 17th problem. (V) Approximation theory: (1) Tougeron's implicit functions theorem; (2) Equivalence of power series; (3) M. Artin's approximation theorem; (4) Formal completion of analytic rings; (5) Nash rings. (VI) Local algebraic rings: (1) Local algebraic rings; (2) Chevalley's theorem; (3) Zariski's main theorem; (4) Normalization and completion; (5) Efroymson's theorem.