### Classification of singularities at infinity of polynomials of degree 4 in two variables.

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For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do...

Corank one mono-germs $f:(\mathbb{R}\u207f,0)\to ({\mathbb{R}}^{p},0)$, n < p, of ${}_{e}$-codimension one are classified by giving an explicit normal form.

We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs $f:({\u2102}^{2},0)\u27f6(\u2102,0)$ which take into account the inflection points of the fibres of $f$. We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.

We implement a singularity theory approach, the path formulation, to classify ${\mathbb{D}}_{3}$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${\mathbb{D}}_{3}$-miniversal unfolding ${F}_{\phantom{\rule{-0.166667em}{0ex}}0}$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of ${F}_{\phantom{\rule{-0.166667em}{0ex}}0}$ onto its unfolding parameter space. We apply our results to degenerate...

Singular projections of generic 2-dim surfaces in ℝ³ with singular boundary to 2-space are studied. The case of projections of surfaces with nonsingular boundary has been treated by Bruce and Giblin. The aim of this paper is to generalise these results to the simplest singular case where the boundary of the surface consists of two transversally intersecting lines. Local models for germs of generic singular projections of corank ≤ 1 and codimension ≤ 3 are given. We also present geometrical realisations...

A classification of simple equivalence classes of function germs with respect to new relations is given. The equivalence relation is similar but weaker than the right action of diffeomorphisms which preserve the boundary. It is used in classifying Lagrange projections with boundary. The simple classes of function germs with respect to the equivalence similar to fibration preserving action are also discussed.

We obtain a complete list of simple framed curve singularities in ℂ² and ℂ³ up to the framed equivalence. We also find all the adjacencies between simple framed curves.

In this paper we study equivalence classes of generic $1$-parameter germs of real analytic families ${\mathcal{Q}}_{\epsilon}$ unfolding codimension $1$ germs of diffeomorphisms ${\mathcal{Q}}_{0}:(\mathbb{R},0)\to (\mathbb{R},0)$ with a fixed point at the origin and multiplier $-1,$ under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, ${\mathcal{P}}_{\epsilon}={\mathcal{Q}}_{\epsilon}^{\circ 2},$ are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle...

We give criteria of finite determinacy for the volume and multiplicities. Given an analytic set described by {v = 0}, we prove that the log-analytic expansion of the volume of the intersection of the set and a "little ball" is determined by that of the set defined by the Taylor expansion of v up to a certain order if the mapping v has an isolated singularity at the origin. We also compare the cardinalities of finite fibers of projections restricted to such a set.