L’objectif dans ce travail est de présenter une généralisation pour l’obstruction d’Euler locale d’une fonction holomorphe singulière à l’origine dans le cas d’une application holomorphe $f:(V,0)\to ({ℂ}^k,0)$, où $(V,0)$ est un germe de variété analytique complexe, équidimensionnel de dimension $n\ge k$. Le résultat principal (Théorème 6.1) exprime l’obstruction d’Euler locale, définie pour un $k$-repère par Brasselet, Seade, Suwa, en fonction de l’obstruction d’Euler relative à $f$.

In this paper we study a notion of local volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study an invariant for normal isolated singularities, generalizing a volume defined by J. Wahl for surfaces. We also compare this generalization to a different one arising in recent work of T. de Fernex, S. Boucksom, and C. Favre.

A mapping $F:ℝⁿ\to {ℝ}^m$ is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping $F:ℝⁿ\to {ℝ}^m$ can be reduced to the case m = n.

Electro-muscular disruption (EMD) devices such as TASER M26 and X26 have been used as a less-than-lethal weapon. Such EMD devices shoot a pair of darts toward an intended target to generate an incapacitating electrical shock. In the use of the EMD device, there have been controversial questions about its safety and effectiveness. To address these questions, we need to investigate the distribution of the current density J inside the target produced by the EMD device. One approach is to develop a computational...

We construct the generic component of the moduli space of the germs of Legendrian curves with generic plane projection topologically equivalent to a curve $y^n=x^m$.

Let f be a germ of plane curve, we define the δ-degree of sufficiency of f to be the smallest integer r such that for anuy germ g such that j(r) f = j(r) g then there is a set of disjoint annuli in S3 whose boundaries consist of a component of the link of f and a component of the link of g. We establish a formula for the δ-degree of sufficiency in terms of link invariants of plane curves singularities and, as a consequence of this formula, we obtain that the δ-degree of sufficiency is equal to the...

Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number ${\varrho }_c\le 1$ such that |x|·|∇f| and ${|f\left(x\right)-c|}^{{\varrho }_c}$ are separated at infinity. If c is a regular value and ${\varrho }_c<1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

Let $M^n$ be a germ at $0\in {ℂ}^m$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi :({ℂ}^n,0)\to (M,0)$ such that ${\phi }^{-1}\left(0\right)=\left\{0\right\}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,...,F_k)$, that is there exists a linear holomorphic vector field $X$ on ${ℂ}^m$, with eigenvalues ${\lambda }_1,...,{\lambda }_m\in {\mathbb{Q}}_{+}$ such that $X\left(F^T\right)=U·F^T$, where $U$ is a $k\times k$ matrix with entries in ${𝒪}_m$. We prove that if there exists...