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Ploski Approximation Theorem

Adam Parusiński, Guillaume Rond

TL;DR

This paper synthesizes Artin's approximation $Ar1$ and Płoski's strengthened approximation to address equisingular deformations and algebrization of analytic and meromorphic germs. It traces constructive methods—from density of convergent solutions to parameterized families $y(x,z)$, via discriminants, Weierstrass preparation, and Zariski equisingularity—to ambient homeomorphisms that render germs algebraic. It also discusses nested generalizations (Nested Artin–Płoski–Popescu) and extends algebrization results to function germs and to two-variable meromorphic germs, highlighting broad applicability. The work strengthens the bridge between analytic and algebraic geometry by providing a cohesive framework for controlled deformations with topological triviality and concrete algebraization of germs.

Abstract

The aim of this paper is to review how some approximation results in commutative algebra are being used to construct equisingular deformations of singularities. The first example of such an approximation result appeared for the first time in A. Ploski's PhD thesis.

Ploski Approximation Theorem

TL;DR

This paper synthesizes Artin's approximation and Płoski's strengthened approximation to address equisingular deformations and algebrization of analytic and meromorphic germs. It traces constructive methods—from density of convergent solutions to parameterized families , via discriminants, Weierstrass preparation, and Zariski equisingularity—to ambient homeomorphisms that render germs algebraic. It also discusses nested generalizations (Nested Artin–Płoski–Popescu) and extends algebrization results to function germs and to two-variable meromorphic germs, highlighting broad applicability. The work strengthens the bridge between analytic and algebraic geometry by providing a cohesive framework for controlled deformations with topological triviality and concrete algebraization of germs.

Abstract

The aim of this paper is to review how some approximation results in commutative algebra are being used to construct equisingular deformations of singularities. The first example of such an approximation result appeared for the first time in A. Ploski's PhD thesis.
Paper Structure (10 sections, 10 theorems, 44 equations)

This paper contains 10 sections, 10 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Approximation Theorems
  3. Artin approximation Theorem
  4. Pł oski approximation Theorem
  5. Algebrization of the germ of an analytic set or of an analytic function
  6. Zariski equisingularity
  7. Construction of a deformation of the germ of an analytic set
  8. Algebraization. Final steps.
  9. Algebraization of function germs.
  10. Algebraization of the germ of a meromorphic function in dimension 2

Key Result

Theorem 2.1

Ar1 Let $x=(x_1,\ldots, x_n)$ and $y=(y_1,\ldots, y_p)$ be an $n$-tuple and a $p$-tuple of indeterminates. Let $f=(f_1,\ldots, f_m)\in\mathbb{C}\{x,y\}^m$ be an $m$-tuple of convergent power series. Assume given a formal power series solution vector $\widehat{y}(x)\in\mathbb{C}\llbracket x\rrbracket Then, for every $c\in\mathbb{N}$, there is a convergent power series solution vector $y(x)\in\mathb

Theorems & Definitions (21)

  • Theorem 2.1: Artin approximation Theorem
  • Theorem 2.2: Pł oski approximation Theorem
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Theorem 2.9: Nested Artin-Pł oski-Popescu Approximation Theorem, BPR
  • Remark 2.10
  • ...and 11 more