Some remarks on the planar Kouchnirenko's theorem

被引：7

作者：

Greuel, Gert-Martin ^{[1
]}

Hong Duc Nguyen ^{[1
,2
]}

机构：

[1] Univ Kaiserslautern, Fachbereich Math, D-67663 Kaiserslautern, Germany

[2] Inst Math, Hanoi 10307, Vietnam

来源：

REVISTA MATEMATICA COMPLUTENSE | 2012年 / 25卷 / 02期

关键词：

Milnor number; Delta-invariant; Newton non-degenerate; Inner Newton non-degenerate; Weak Newton non-degenerate; MILNOR NUMBERS; CURVES; SMOOTH;

D O I：

10.1007/s13163-011-0082-7

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities {f(x, y) = 0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number mu resp. the delta-invariant delta can be computed by explicit formulas mu(N) resp. delta(N) from the Newton diagram of f if f is NND resp. WNND. It was however unknown whether the equalities mu = mu(N) resp. delta = delta(N) can be characterized by a certain non-degeneracy condition on f and, if so, by which one. We show that mu = mu(N) resp. delta = delta(N) is equivalent to INND resp. WHNND and give some applications and interesting examples related to the existence of "wild vanishing cycles". Although the results are new in any characteristic, the main difficulties arise in positive characteristic.

引用

页码：557 / 579

页数：23

共 13 条

[1] The Newton polygon of plane curves with many rational points [J].