Approximation of sets defined by polynomials with holomorphic coefficients

TL;DR

This paper establishes conditions under which sets defined by polynomials with holomorphic coefficients can be approximated by sequences of similar sets, ensuring convergence of the sets as the coefficients vary holomorphically.

Contribution

It provides a framework for approximating analytic sets defined by polynomials with holomorphic coefficients through converging sequences of coefficients.

Findings

Conditions for convergence of polynomial-defined sets with holomorphic coefficients

Sequences of approximating sets converge to the original set under specified conditions

Framework applicable to complex analytic geometry and approximation theory

Abstract

Let X be an analytic set defined by polynomials whose coefficients a_1,...,a_s are holomorphic functions. We formulate conditions such that for all sequences {a_(1,n)},...,{a_(s,n)} of holomorphic functions converging locally uniformly to a_1,...,a_s respectively the following holds true. If a_(1,n),...,a_(s,n) satisfy the conditions then the sequence of the sets {X_n} obtained by replacing a_j by a_(j,n) in the polynomials, converge to X.