Motivic invariant of real polynomial functions and Newton polyhedron

We propose a computation of real motivic zeta functions for real polynomial functions, using Newton polyhedron. As a consequence we show that the weights are blow-Nash invariants of convenient weighted homogeneous polynomials in three variables.Comment: 22 pages in Math. Proc. Camb. Phil. Soc, 201

Singularities of parallel surfaces

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularities types in terms of differential geometry (Theorem 3.4 and 3.5).Comment: 24 pages, final version, to appear in Tohoku Mathematical Journa

On the topology of stable maps

International audienceWe investigate how Viro's integral calculus applies for the study of the topology of stable maps. We also discuss several applications to Morin maps and complex maps

Versality of Rotation Unfolding of Folding Maps for Surfaces in R3\mathbb{R}^3

We introduce the rotation unfolding of the folding map of a surface in $\mathbb{R}^3$, and investigate its $\mathcal{A}$-vesality. The rotation unfolding is a 2-parameter unfolding and can be considered as a subfamily of the folding family, which is introduced by Bruce and Wilkinson. They revealed relationships between a bifurcation set of this family and the focal/symmetry set of a surface in $\mathbb{R}^3$. We state the criteria of singularities of the folding map up to codimension 2 and prove when our rotation unfolding is versal. The conditions to be versal are stated in terms of geometry. As a by-product, we show the diffeomorphic type of the locus of the tangent planes of the focal set of regular surfaces, which passes through the origin.Comment: 13 pege, 6 figures, Extension of the Singularity theor