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On the equisingularity class of the general higher order polars of plane branches

Evelia R. García Barroso, Janusz Gwoździewicz, Mateusz Masternak

Abstract

In this paper we describe the factorization of the higher order polars of a generic branch in its equisingularity class. We generalize the results of Casas-Alvero and Hefez-Hernandes-Hernández to higher order polars.

On the equisingularity class of the general higher order polars of plane branches

Abstract

In this paper we describe the factorization of the higher order polars of a generic branch in its equisingularity class. We generalize the results of Casas-Alvero and Hefez-Hernandes-Hernández to higher order polars.

Paper Structure

This paper contains 4 sections, 7 theorems, 51 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Symbolic derivatives of a Newton diagram
  3. Technical tools
  4. Proof of the main theorem

Key Result

Theorem 1.1

Let $f\in \mathbb C[[x,y]]$ be a generic element of $K(b_0,\ldots,b_h)$. Put $e_{i} =\gcd(b_{0}, \ldots , b_{i})$, $n_{i}=\frac{e_{i-1}}{e_{i}}$, $m_{i}=\frac{b_{i}}{e_{i}}$ and $\Delta_{i}=\Bigl\{ \begin{picture}(4,3)(0,0.4) \put(0,1.15){\line(1,0){4}} \put(0,0.85){\line(1,0){4}} where, for any $\ell\in \{1,\ldots,i_{k}\}$, the power series $\Gamma^{(\ell)}$ is not necessarily

Figures (8)

  • Figure 1: Elementary Newton diagram
  • Figure 2: Canonical and long canonical representation of $\Bigl\{ \Bigr\}+\Bigl\{ \Bigr\}$
  • Figure 3: Symbolic derivatives
  • Figure 4: Points $A_{i-1}$ and $A_i$
  • Figure 5: Points $B_i$
  • ...and 3 more figures

Theorems & Definitions (21)

  • Example 1
  • Example 2
  • Theorem 1.1
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Proposition 2.7
  • proof
  • ...and 11 more