Related papers: Determinantal singularities and Newton polyhedra

Mather and Yau showed that an isolated complex hypersurface singularity is completely determined by its moduli algebra. It is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and,…

All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as…

We study a class of holomorphic $p$-forms satisfying nondegeneracy conditions expressed through their Newton polyhedron and called Newton nondegenerate (NND). We give a characterization of NND $p$-forms by their toric reduction of…

A linear map between two vector spaces has a very important characteristic: a determinant. In modern theory two generalizations of linear maps are intensively used: to linear complexes (the nilpotent chains of linear maps) and to non-linear…

We give an effective method to determine the multiplier ideals and jumping numbers associated with a curve singularity $C$ in a smooth surface. We characterize the multiplier ideals in terms of certain Newton polygons, generalizing a…

The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…

We define the toric Newton spectrum of a polynomial and we give some applications in singularity theory, combinatorics and mirror symmetry.

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for…

We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal…

We discuss the distribution of the spectrum at infinity of a convenient and nondegenerate Laurent polynomial (singularity side) and the distribution of the Newton spectrum of a polytope (Ehrhart theory side). To this end, we study a hard…

In this paper, the discriminant of homogeneous polynomials is studied in two particular cases: a single homogeneous polynomial and a collection of n-1 homogeneous polynomials in n variables. In these two cases, the discriminant is defined…

Some changes in a recent convolution formula are performed here in order to clean it up by using more conventional notations and by making use of more referrenced and documented components (namely Sierpi\'nski's polynomials, the Thue-Morse…

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function).…

The contents are divided into two papers "The Monotone Cumulants" (arXiv:0907.4896) and "Conditionally monotone independence" (arXiv:0907.5473).

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this…

In the study of normal surface singularities the relation between analytical and topological properties and invariants of the singularity is a very rich problem. This relation is particularly close for surface singularities constructed from…

First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function…

Let $ \mathcal{A}_1, \ldots, \mathcal{A}_k $ be finite sets in $ \mathbb{Z}^n $ and let $ Y \subset (\mathbb{C}^*)^n $ be an algebraic variety defined by a system of equations \[ f_1 = \ldots = f_k = 0, \] where $ f_1, \ldots, f_k $ are…

Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties.For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear…