Approximation of Analytic Sets with Proper Projection by Algebraic Sets

Abstract

Let X be an analytic subset of U × C n of pure dimension k such that the projection of X onto U is a proper mapping, where U ⊂ C k is a Runge domain. We show that X can be approximated by algebraic sets. Next we present a constructive method for local approximation of analytic sets by algebraic ones.

Effective approximation of the solutions of algebraic equations

“…Then one cannot simply replace the defining functions by approximating polynomials as this gives sets of strictly smaller dimensions. Nevertheless, algebraic approximations do exist for a large subclass of the class of non-complete intersections (see [5], [6], [8]). In particular, every analytic set admits local algebraic approximations which in many cases can be effectively constructed.…”

“…One of the main tools used in the construction is Theorem 3.1 discussed in the present paper (cf. [6], p. 284) and the algorithm following from its proof.…”

“…Consequently, P ν 1 , P ν 6 corresponding to the Nash approximations f ν 1 , f ν 6 of f 1 , f 6 , respectively, will not be monic either but their leading coefficients will be non-vanishing at 0 and their degrees in z 1 , z 6 , respectively, will be independent of ν, as required in (b). As before, we are allowed to use only the data representing A, a i , b i described in Section 3.2.1 (obtained by applying the effective versions of the Weierstrass Preparation and Division Theorems in Steps 5,6) and the equations defining the variety (Step 7).…”

“…To do this, we compute the order κ of zero of δ(x 1 ) := Q1 (x 1 )A(x 1 ) + Q2 (x 1 ) at 0 to confirm that κ = 6. Since (after Steps 5, 6) we have Expand ai , Expand bi , Expand A , M ai , M bi , M A , we also have Expand δ(x1)/x 6 1 , M δ(x1)/x 6 1 (where U δ(x1)/x 6 1 = D 1 ). Therefore, we can check that inf D 1 3 |δ(x 1 )/x 6 1 | > 0 which implies that 0 is the only root of δ in D 1 3 .…”

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.…”