Characterization of uni-modal parametric plane curve singularities

Abstract

Communicated by B. BuchbergerIn this article we characterize the classification of uni-modal parametric plane curve singularities given by Ishikawa and Janeczko, in terms of invariants. On the basis of this characterization we present an algorithm to classify the uni-modal parametric plane 1750039-1 J. Algebra Appl. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 06/26/16. For personal use only. M. A. Binyamin et al.curve singularities and also give its implementation in computer algebra system SIN-GULAR.

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

“…A consequence is that the results of the classification in characteristic 0 hold in characteristic p if p > c(Γ). A classification of the unimodal parametric plane branches in characteristic 0 was first given in [6], which we recall in Table 5. The classification uses theorem 7.1 and therefore hold also for p > c(Γ).…”

“…There are several further contributions in characteristic zero (cf. [6], [18], [19], [21], [24], [26], [27]). In [19], Hefez and Hernandes introduced a general approach using differential forms for the classification of complex analytic parametric plane curve singularities.…”

“…Using values of differentials, they derived a "very short" parametrization, improving Zariski's short parametrization, but without giving explicit lists. In [27] Mehmood and Pfister used another approach to classify the simple and unimodal parametric plane branches in characteristic 0 and gave explicit lists (see also [6] for a classifier). They also classified simple parametric plane curve singularities in positive characteristic, see [28].…”

“…Unimodal Parametric Plane Curve Singularities in Characteristic p, p = 0[6],[27] or p > c(Γ) (1 series, 28 sporadic singularities, a ∈ K). , t9 + t 10 + at 11 ), a = 19 18 10 11 (t 4 , t 9 + t 10 + 19 18 t 11 + at 15 ) 10 15 (t 4 , t 9 + t 11 ) , t 10 + t 2k+1 at 2k+3 ) 4, 10, 2k + 11 2k+1 2k+3 (t 4 , t 11 + t 13 + at 14 ) 13 14 (t 4 , t 11 + t 14 + at 17 ), a = 25 22 14 17 (t 4 , t 11 + t 14 + 25 22 t 17 + at 21 ) 14 21 (t 4 , t 11 + t 17 ) 4, 11 17 -(t 4 , t 11 + t 21 ) 21 -(t 4 , t 11 + t 25 ) 25 -(t 4 , t 11 ) ∞ -(t 5 , t 6 + t 8 + at 9 ) 8 9 (t 5 , t 6 + t 9 ) , t 7 + t 8 + at 11 ) 8 11 (t 5 , t 7 + t 11 + at 13 ) , t 8 + t 9 + at 12 ) 9 12 (t 5 , t 8 + t 12 + at 14 ) 5, 8 12 14 (t 5 , t 8 + t 14 + at 17 ) 14 17 (t 5 , t 8 + t 17 ) 17 -(t 5 , t 8 + t 22 ) 22 -(t 5 , t 8 ) ∞ -Lemma 7.2.…”