scholarly journals Polar cremona transformations and monodromy of polynomials

2010 ◽  
Vol 47 (1) ◽  
pp. 81-89
Author(s):  
Imran Ahmed
Keyword(s):  
Lower Bound ◽  
Homogeneous Polynomial ◽  
Isolated Singularities ◽  
Cremona Transformations ◽  
Projective Hypersurface ◽  
Open Set

Consider the gradient map associated to any non-constant homogeneous polynomial f ∈ ℂ[ x0 , ..., xn ] of degree d , defined by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\varphi _f = grad (f):D(f) \to \mathbb{P}^n ,(x_0 : \ldots :x_n ) \to (f_0 (x): \ldots :f_n (x))$$ \end{document} where D ( f ) = { x ∈ ℙ n ; f ( x ) ≠ 0} is the principal open set associated to f and fi = ∂f / ∂xi . This map corresponds to polar Cremona transformations. In Proposition 3.4 we give a new lower bound for the degree d ( f ) of ϕ f under the assumption that the projective hypersurface V : f = 0 has only isolated singularities. When d ( f ) = 1, Theorem 4.2 yields very strong conditions on the singularities of V .

scholarly journals Identifiability of homogeneous polynomials and Cremona transformations

2019 ◽  
Vol 2019 (757) ◽  
pp. 279-308 ◽  
Author(s):  
Francesco Galuppi ◽  
Massimiliano Mella
Keyword(s):  
Special Class ◽  
Homogeneous Polynomial ◽  
Projective Spaces ◽  
Homogeneous Polynomials ◽  
Additive Decomposition ◽  
Linear Forms ◽  
Cremona Transformations ◽  
General Polynomial ◽  
Powers Of Linear Forms

AbstractA homogeneous polynomial of degree d in {n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in {n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

scholarly journals Remarks on the boundary set of spectral equipartitions

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120492 ◽  
Author(s):  
P. Bérard ◽  
B. Helffer
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Two Dimensional ◽  
Nodal Sets ◽  
Nodal Domains ◽  
Open Set ◽  
Ground State Energies

Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ ( ω j ) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

Heat equation and the principle of not feeling the boundary

1989 ◽  
Vol 112 (3-4) ◽  
pp. 257-262 ◽  
Author(s):  
M. van den Berg
Keyword(s):  
Lower Bound ◽  
Heat Equation ◽  
Heat Kernel ◽  
Open Set ◽  
Dirichlet Heat Kernel

SynopsisWe prove a lower bound for the Dirichlet heat kernel pD(x,y;t), where x and y are a visible pair of points in an open set D in ℝm.

POLYNOMIAL VERSION ON THE SUM-PRODUCT PROBLEM

2020 ◽  
Vol 2020 (2) ◽  
pp. 4-10
Author(s):  
Sofia Aleshina ◽  
Ilya Vyugin
Keyword(s):  
Lower Bound ◽  
General Principle ◽  
Homogeneous Polynomial ◽  
Minkowski Sum ◽  
Theory Of Information ◽  
Product Problem

This work is about the generalization of sum-product problem. The general principle of it was formulated in the Erdos-Szemeredi’s hypothesis. Instead of the Minkowski sum in this hypothesis, the set of values f(x,y) of a homogeneous polynomial f lin two variables, where x and y belong to subgroup G of is considered. The lower bound on the cardinality of such set is obtained. This topic has an applied value in the theory of information and dynamics in calculating the probabilities of events, as well as in various methods of encoding and decoding information.

scholarly journals Koszul Complexes and Pole Order Filtrations

2014 ◽  
Vol 58 (2) ◽  
pp. 333-354 ◽  
Author(s):  
Alexandru Dimca ◽  
Gabriel Sticlaru
Keyword(s):  
Linear Systems ◽  
Homogeneous Polynomial ◽  
The Other ◽  
Koszul Complex ◽  
Partial Derivatives ◽  
Nodal Surface ◽  
Pole Order ◽  
Open Set ◽  
The One ◽  
Derivatives Of

AbstractWe study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial f and the pole order filtration P on the cohomology of the open set U = ℙn \ D, with D the hypersurface defined by f = 0. The relation is expressed by some spectral sequences. These sequences may, on the one hand, in many cases be used to determine the filtration P for curves and surfaces and, on the other hand, to obtain information about the syzygies involving the partial derivatives of the polynomial f. The case of a nodal hypersurface D is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of D. When D is a nodal surface in ℙ3, we show that F2H3(U) ≠ P2H3(U) as soon as the degree of D is at least 4.

scholarly journals Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

2017 ◽  
Vol 28 (05) ◽  
pp. 1750029 ◽  
Author(s):  
Christophe Eyral
Keyword(s):  
Special Class ◽  
Homogeneous Polynomial ◽  
Positive Answer ◽  
Isolated Singularities ◽  
Hypersurface Singularities ◽  
Line Singularities

We prove that if [Formula: see text] is a family of line singularities with constant Lê numbers and such that [Formula: see text] is a homogeneous polynomial, then [Formula: see text] is equimultiple. This extends to line singularities a well-known theorem of Gabrièlov and Kušnirenko concerning isolated singularities. As an application, we show that if [Formula: see text] is a topologically [Formula: see text]-equisingular family of line singularities, with [Formula: see text] homogeneous, then [Formula: see text] is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.

scholarly journals Foliations with isolated singularities on Hirzebruch surfaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos Galindo ◽  
Francisco Monserrat ◽  
Jorge Olivares
Keyword(s):  
Projective Plane ◽  
Tangent Bundle ◽  
Homogeneous Polynomial ◽  
Isolated Singularities ◽  
Hirzebruch Surfaces

Abstract We study foliations ℱ {\mathcal{F}} on Hirzebruch surfaces S δ {S_{\delta}} and prove that, similarly to those on the projective plane, any ℱ {\mathcal{F}} can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ {\mathcal{F}} has isolated singularities, we show that, for δ = 1 {\delta=1} , the singular scheme of ℱ {\mathcal{F}} does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ ≠ 1 {\delta\neq 1} , we prove that the singular scheme of ℱ {\mathcal{F}} does not determine the foliation. However, we prove that, in most cases, two foliations ℱ {\mathcal{F}} and ℱ ′ {\mathcal{F}^{\prime}} given by sections s and s ′ {s^{\prime}} have the same singular scheme if and only if s ′ = Φ ⁢ ( s ) {s^{\prime}=\Phi(s)} , for some global endomorphism Φ of the tangent bundle of S δ {S_{\delta}} .

Some effectivity questions for plane Cremona transformations in the context of symmetric key cryptography

2021 ◽  
Vol 64 (1) ◽  
pp. 1-28
Author(s):  
N. I. Shepherd-Barron
Keyword(s):  
Lower Bound ◽  
Hyperbolic Plane ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Henon Maps ◽  
Hénon Maps ◽  
Symmetric Key ◽  
Symmetric Key Cryptography ◽  
Feistel Ciphers ◽  
Plane Cremona Transformations

An effective lower bound on the entropy of some explicit quadratic plane Cremona transformations is given. The motivation is that such transformations (Hénon maps, or Feistel ciphers) are used in symmetric key cryptography. Moreover, a hyperbolic plane Cremona transformation g is rigid, in the sense of [5], and under further explicit conditions some power of g is tight.

scholarly journals On the volume of isolated singularities

2014 ◽  
Vol 150 (8) ◽  
pp. 1413-1424 ◽  
Author(s):  
Yuchen Zhang
Keyword(s):  
Lower Bound ◽  
Duke Math ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Local Volume ◽  
Equivalent Definition ◽  
Log Canonical ◽  
Definition Of

AbstractWe give an equivalent definition of the local volume of an isolated singularity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\rm Vol}_{\text {BdFF}}(X,0)$ given in [S. Boucksom, T. de Fernex, C. Favre, The volume of an isolated singularity. Duke Math. J. 161 (2012), 1455–1520] in the $\mathbb{Q}$-Gorenstein case and we generalize it to the non-$\mathbb{Q}$-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if $X$ is Gorenstein. We also give a non-$\mathbb{Q}$-Gorenstein example with ${\rm Vol}_{\text {BdFF}}(X,0)=0$, which does not allow a boundary $\Delta $ such that the pair $(X,\Delta )$ is log canonical.

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