Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one
Andrzej Lenarcik, Mateusz Masternak · Wydawnictwo Uniwersytetu Łódzkiego eBooks · 2022
Let \( f \in \mathbb{C}\{X, Y\} \) be a reduced series which defines a singular branch \( f = 0 \) in a neighbourhood of zero in \( \mathbb{C}^2 \). Let \( h \geq 1 \) be the number of characteristic exponents of a Puiseux root \( y(X) \in \mathbb{C}\{X\}^* \) of the equation \( f = 0 \). For any \( k \in \{1, \ldots, h\} \) we define the series \( f_k \in \mathbb{C}\{X, Y\} \) generated by all terms of the series \( y(X) \) with orders strictly smaller than the \( k \)-th characteristic exponent. We consider a deformation \( F_t = f + tX^{\omega_0} f_1^{\omega_1} \ldots f_h^{\omega_h} \) (\( t eq 0 \), small) where \( \omega_0, \omega_1, \ldots, \omega_h \) are nonnegative integers. Using a version of the Newton algorithm proposed by Cano we show how to choose exponents \( \omega_0, \omega_1, \ldots, \omega_h \) to obtain the Milnor number of the deformation \( F_t \) smaller by one than the Milnor number of the branch \( f \). We prove a version of Kouchnirenko theorem which is useful in computation of the Milnor number.