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We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in C ∞ (N × [ 0 , 1 ] , P) is dense if and only if the pair of dimensions (dim N , dim P) is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs (n, p) for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have A e -codimension 1 hyperplane sections. They are also related to the simplicity of A e -codimension 2 germs. We give a sufficient condition for any A e -codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if $(X,0)\subset (\mathbb {C}^{n},0)$ , $(Y,0)\subset (\mathbb {C}^{m},0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: if $X\subset \mathbb {C}^{n}$ , $Y\subset \mathbb {C}^{m}$ are algebraic sets and $\phi \colon X\to Y$ is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb {P}^{n+m}(\mathbb {C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$. [ABSTRACT FROM AUTHOR]

In this paper, a systematic method is given to construct all liftable vector fields over an analytic multigerm $$f:({\mathbb {K}}^n, S)\rightarrow ({\mathbb {K}}^p,0)$$ of corank at most one admitting a one-parameter stable unfolding. [ABSTRACT FROM AUTHOR]

We study classes of mappings which do not belong to the QRT family. We obtain several integrable non-autonomous forms of these mappings extending previous results where only linearisable cases were found. Using our recently introduced method of singularity confinement with full deautonomisation, we analyse a mapping which, while non-integrable, does possess confined singularities and show that our method makes it possible to obtain the exact value of its algebraic entropy. [ABSTRACT FROM AUTHOR]

This paper relates the multiple point spaces in the source and target of a corank 1 map-germ $${(\mathbb {C}^n, 0)\to(\mathbb {C}^{n+1}, 0)}$$ . Let f be such a map-germ, and, for 1 ≤ k ≤ multiplicity( f ), let D( f ) be its k'th multiple point scheme - the closure of the set of ordered k-tuples of pairwise distinct points sharing the same image. There are natural projections D( f ) → D( f ), determined by forgetting one member of the ( k + 1)-tuple. We prove that the matrix of a presentation of $${\mathcal {O}_{D^{k+1}(f)}}$$ over $${\mathcal {O}_{D^k(f)}}$$ appears as a certain submatrix of the matrix of a suitable presentation of $${\mathcal {O}_{\mathbb {C}^n,0}}$$ over $${\mathcal {O}_{\mathbb {C}^{n+1},0}}$$ . This does not happen for germs of corank > 1. [ABSTRACT FROM AUTHOR]

We give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions. [ABSTRACT FROM PUBLISHER]

In this paper, a criterion on the C 0-sufficiency for a function germ with non-isolated singularity is obtained analogously to that of Kuiper–Kuo for the case of isolated singularities. Moreover, the Kuiper–Kuo condition and the Thom condition for an analytic function germ with non-isolated singularity are proved to be equivalent. [ABSTRACT FROM AUTHOR]

Risler & Trotman in 1997 proved that the multiplicity of an analytic function germ is left–right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman & Trotman, we generalize the work of them to prove that the multiplicity of a C ∞–differentiable function germ is also left–right lipschitz invariant. [ABSTRACT FROM AUTHOR]