Indices of Newton non-degenerate vector fields and a conjecture of Loewner for surfaces in R 4 :

We study the index of a vector field inR, with isolated singularity, in terms of conditions on the Newton polyhedra associated to its coordinates. When the vector field is Newton non-degenerate, we show that its index is determined by the principal part of the Newton polyhedra. As a consequence we can prove that, under very mild conditions, the index of an isolated inflection point of a locally convex surface generically embedded in R is the same as the index of an umbilic point of a surface immersed in R. October, 2001 ICMC-USP