Comments about Hilbert's 16'th problem

TL;DR

This paper discusses conditions under which local analytic germs can be deformed in a way that respects flow symmetry, focusing on the existence of a high-degree holomorphic solution relative to local discrepancy.

Contribution

It introduces a criterion linking the existence of a high-degree holomorphic solution to the equivariant deformation of local analytic germs.

Findings

High-degree holomorphic solutions facilitate equivariant deformations.

A relation between solution degree and local discrepancy is established.

Implications for understanding deformations in complex analytic settings.

Abstract

Local analytic germs can be simultaneously deformed equivariantly for the flow if there is one holomorphic solution whose degree is high compared to local discrepancy.