Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one - Lenarcik, Andrzej; Masternak, Mateusz (Analitic and Algebraic Geometry 4, 2022)
Let \( f C{X, Y} \) be a reduced series which defines a singular branch \( f = 0 \) in a neighbourhood of zero in \( C^2 \). Let \( h ≥ 1 \) be the number of characteristic exponents of a Puiseux root \( y(X) C{X}^* \) of the equation \( f = 0 \). For any \( k {1, …, h} \) we define the series \( f_k C{X, Y} \) generated by all terms of the series \( y(X) \) with orders strictly smaller than the \( k \)-th characteristic exponent. We consider a deformation \( F_t = f + tX^ω_0 f_1^ω_1 … f_h^ω_h \) (\( t eq 0 \), small) where \( ω_0, ω_1, …, ω_h \) are nonnegative integers. Using a version of the Newton algorithm proposed by Cano we show how to choose exponents \( ω_0, ω_1, …, ω_h \) 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.