Related papers: The Vandermonde determinant identity in higher dim…
The discrete polymatroids and their base rings are studied recently in many papers (see \cite{HH}, \cite{HHV}, \cite{V1}, \cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
Using vector fields we obtain an irreducibility criterion for hypersurfaces. It only requires the Weierstrass division.
In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of…
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
In the spirit of Bar Natan's construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\vec{x}=(x_1,...,x_n)$, we construct a commutative diagram in the shape of the…
Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for…
We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure and we prove that it is…
In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension $d$ and order $h$ with a generic differential hypersurface of order $s$ is shown…
A classical result of Boole shows that, in characteristic 0, the set of singular degree d hypersurfaces in P^N is a divisor of degree (N+1)(d-1)^N in the projective space of all hypersurfaces. We give here analogous formulae for complete…
Given two varieties V,W in the n-fold product of modular curves, we answer affirmatively a question (formulated by Shou-Wu Zhang's AIM group) on whether the set of points in V that are Hecke translations of some point on W is dense in V. We…
A point P on a smooth hypersurface X of degree d in an N-dimensional projective space is called a star point if and only if the intersection of X with the embedded tangent space T_P(X) is a cone with vertex P. This notion is a…
We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures.…
In this paper we prove some combinatorial identities which can be considered as generalizations and variations of remarkable Chu-Vandermonde identity. These identities are proved by using an elementary combinatorial-probabilistic approach…
In this paper, we establish a second main theorem for holomorphic curve intersecting hypersurfaces in general position in projective space with level of truncation. As an application, we reduce the number hypersurfaces in uniqueness problem…
Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call {\em Vandermonde varieties}. They are closely related to the varieties consisting of all…
A conjecture for higher order separation on generic rational surfaces with some new results about standard divisors.
We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian…
We study a generalisation of the concept of hyperuniformity to spheres of arbitrary dimension. It is shown that QMC-designs (and especially spherical designs) are hyperuniform in our sense.