  Let $S$ be a Puiseux series of the germ of an analytically irreducible plane
curve $Z$. We provide a new perspective to construct a set of polynomials
$F=\{F_1,\ldots, F_{g-1}\}$ associated to $S$, which is a special choice of
maximal contact elements constructed in [AMB17] and approximate roots defined
in [Dur18], [AM73a], [AM73b]. Using these polynomials as building blocks, we
describe a set of generators of multiplier ideals of the form
$\mathfrak{I}(\alpha Z)$ with $0<\alpha<1$ a rational number, which recovers
the results about irreducible plane curves in [AMB17], [Dur18].
