The right classification of univariate power series in positive characteristic

Nguyen Hong Duc

Journal of Singularities
volume 10 (2014), 235-249

Received 3 February 2013. Received in revised form 16 January 2014.

DOI: 10.5427/jsing.2014.10p

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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.

Author(s) information:

Nguyen Hong Duc
Institute of Mathematics
18 Hoang Quoc Viet Road
Cau Giay District 10307, Hanoi, Vietnam