Related papers: Green's Hyperplane Restriction Theorem: an extensi…

For each $n$, let RD$(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In this paper, we recover an algorithm of Sylvester for determining…

We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in $\mathbb{P}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and…

We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an…

For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$…

Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas…

New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of…

We obtain a non--trivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to growth of conjugates classes, estimates of…

We prove a strengthening of the Grothendieck-Lefschetz hyperplane theorem for local Picard groups conjectured by Koll\'ar. Our approach, which relies on acyclicity results for absolute integral closures, also leads to a restriction theorem…

Let $R=\bigoplus_{n\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\dots,x_e$ of positive degrees $d_1,\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module…

We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of $\mathbb C^n$, in the case where poles collide. We consider the case where all poles do not converge to the same point in the…

The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…

In 2006, in a paper published in Compositio titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer derived bounds for the canonical Green's function and the hyperbolic Green's function defined on a compact hyperbolic…

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of…

We study closedness of the range, adjointability and generalized invertibility of modular operators between Hilbert modules over locally C*-algebras of coefficients. Our investigations and the recent results of M. Frank [Characterizing…

We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.

In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction.

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…

In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the…

Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential…

Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…