On the relative Nash approximation of analytic maps

Abstract

Let X, Y be real or coxnplex Nash spaces aríd Z a subspace of X; the paper deals with the approxirnation of analytic niaps X -* Y, Nash orí Z, with Nash maps 4' : X -* Y such that 4'Iz =~Iz.

Approximation of analytic sets with proper projection by Nash sets

“…The problem of algebraic approximation of analytic objects appears naturally in complex geometry and has been considered by several mathematicians (see [5,[7][8][9][10]). Various results have been obtained, especially when approximated objects are holomorphic mappings or manifolds.…”

Approximation of holomorphic maps from Runge domains to affine algebraic varieties

“…The original proof of Lempert's approximation theorem [21], pp 338-339, relies on the general Néron desingularization, a deep and difficult result of commutative algebra for which the reader is referred to [1], [26], [27], [28], [29], [30]. Theorem 1.1 is expressed in terms of analytic geometry and has had numerous applications in the theory of several complex variables (see [5], [11], [12], [13], [21], [23], [25], [31]). It is natural to ask whether one can replace Néron desingularization by simpler geometric methods.…”

Approximation of analytic sets by Nash tangents of higher order

“…The idea of algebraic approximation of analytic objects in complex geometry has been developed by several mathematicians (see [3][4][5]9,11,[17][18][19]). In this paper we investigate the approximation of complex analytic sets by complex Nash sets which is expressed in terms of the convergence of holomorphic chains.…”