Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

Abstract

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 exponets 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 Ft = f + tX ω 0 f ω 1 1 . . . f ω h h (t ̸ = 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 Ft smaller by one than the Milnor number of the branch f . We prove a version of Kouchnirenko theorem which is useful in computation the Milnor number.