On the analytic classification of irreducible plane curve singularities (Q2152626)

The most common way of classifying complex analytic plane curve singularities is via equisingularity. This is a topological classification, but there is another classification, the analytic one. If one considers irreducible singularities, the Puiseux series of an irreducible germ of plane curve \(\gamma\) is an expression of the form \(y=h(x)\), where \(h(x)\) is a convergent fractionary power series and \(\{x,y\}\) suitable local coordinates. Some of the exponents of \(h(x)\) are named characteristic exponents and they determine the equisingularity class of \(\gamma\). However, the analytic type also depends of some of the coefficients in \(h(x)\). Moduli spaces where introduced by Zariski to parametrize analytic classes of irreducible singularities with fixed equisingularity class and most of the researching on analytic classification follows this line. This interesting paper addresses the problem of determining those coefficients of \(h(x)\) that affect the analytic type of an irreducible germ of plane curve in the case of having only one characteristic exponent. Notice that there are finitely many coefficients of this type. This is a hard problem since changing a coefficient may or not affect the analytic type depending on the former ones. The author introduces a new approach. He considers an equation of the germ and associates a coefficient of \(h(x)\) to each coefficient of the equation giving rise to pairs which satisfy that the variation of one of them causes a variation of the analytic type if and only if this happens with the variation of the other. When this holds, these coefficients are called relevant. Considering a particular type of equation of the germ, the sub-indices of their non-vanishing coefficients are considered in the Newton plane. Taking the Zariski point, certain line through this point and other parallel lines, appear certain regions in the Newton plane, named of critical coefficients, continuous invariants or conditional invariants, respectively. The main result of the paper is to prove that these regions contain all relevant coefficients and the points in those regions have additional properties explained in the paper. This new approach also allows the author to recover many known results on the subject.