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We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant…
We study the Hamiltonian vector field $v=(-\partial f/\partial w,\partial f/\partial z)$ on $\mathbb C^2$, where $f=f(z,w)$ is a polynomial in two complex variables, which is non-degenerate with respect to its Newton's polygon. We introduce…
In this paper, we establish a uniqueness theorem for algebraically nondegenerate meromorphic maps of C^m into C P^n and slowly moving hypersurfaces Q_j in C P^n, j=1,...,q in (weakly) general position, where q depends effectively on n and…
Given a multivariate generating function F, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of F, resulting in a tractable oscillating integral. This paper treats the case…
This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi…
The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions,…
This paper studies a large class of continuous functions $f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes such classical examples as…
A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we…
We define a nonnegative integer $\la(L,L_0;\phi)$ for a pair of diffeomorphic closed Lagrangian surfaces $L_0,L$ embedded in a symplectic 4-manifold $(M,\w)$ and a diffeomorphism $\phi\in\Diff^+(M)$ satisfying $\phi(L_0)=L$. We prove that…
We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of…
Under the generalized Lindel\"{o}f hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.
We demonstrate that the failure of $L^1$ regularity in Calder\'on-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal,…
We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…
Consider the gradient map associated to any non-constant homogeneous polynomial $f\in \C[x_0,...,x_n]$ of degree $d$, defined by \[\phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x))\] where $D(f)=\{x\in \CP^n; f(x)\neq 0\}$…
The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities.…
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of…
We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…
We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.
A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant…
We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$…