Journal of Singularities
volume 10 (2014), 235-249
Received 3 February 2013. Received in revised form 16 January 2014.
While the classification of univariate power series up to coordinate change is trivial in characteristic 0, this classification is very different in positive characteristic. In this note we give a complete classification of univariate power series in K[[x]], where K is an algebraically closed field of characteristic p>0 by explicit normal forms. We show that the right determinacy of f is completely determined by its support. Moreover we prove that the right modality of f is equal to the integer part of \mu/p, where \mu is the Milnor number of f. As a consequence we prove in this case that the modality is equal to the proper modality, which is the dimension of the \mu-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section.
Nguyen Hong Duc
Institute of Mathematics
18 Hoang Quoc Viet Road
Cau Giay District 10307, Hanoi, Vietnam