On polynomials depending on coefficients of Puiseux parametrizations

Abstract

Every irreducible germ of singular complex plane analytic curve at the origin can be described in two ways in a given coordinate system: by a parametrization $x=T^n$, $y={\sum }_{i=1}^{\infty }{a}_i{T}^i$ or by an equation $f(x,y)=0$ where $f=\sum c_{ij}{X}^i{Y}^j$ is a complex power series in two variables. We show that every polynomial condition on the coefficients of a parametrization is, under some natural invariance assumptions, equivalent with a polynomial condition on the coefficients of f.