Algebraic approximation of analytic sets and mappings

TL;DR

This paper introduces a condition for the convergence of analytic sets and applies it to develop an algorithm that approximates holomorphic solutions of Nash equations by Nash solutions.

Contribution

It presents a new condition ensuring component-wise convergence of analytic sets and uses it to create an algorithm for approximating holomorphic solutions of Nash systems.

Findings

Established a convergence condition for irreducible components of analytic sets.

Developed an algorithm for approximating holomorphic solutions by Nash solutions.

Demonstrated the applicability of the method to systems of Nash equations.

Abstract

Let {X_n} be a sequence of analytic sets converging to some analytic set X in the sense of holomorphic chains. We introduce a condition which implies that every irreducible component of X is the limit of a sequence of irreducible components of the sets from {X_n}. Next we apply the condition to approximate a holomorphic solution y=f(x) of a system Q(x,y)=0 of Nash equations by Nash solutions. Presented methods allow to construct an algorithm of approximation of the holomorphic solutions.