Singularities of k-tuples of vector fields

Bronisław Jakubczyk; Feliks Przytycki

Displaying similar documents to “Singularities of k-tuples of vector fields”

On the classification of real mono-germs of corank one and codimension one

Kevin Houston (2004)

Banach Center Publications

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Corank one mono-germs $f : (ℝ ⁿ, 0) \to (ℝ^{p}, 0)$ , n < p, of $_{e}$ -codimension one are classified by giving an explicit normal form.

Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)

Bulletin de la Société Mathématique de France

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We establish in this paper a lower bound for the volume of a unit vector field $\vec{v}$ defined on $𝐒^{n} ∖ {\pm x}$ , $n = 2, 3$ . This lower bound is related to the sum of the absolute values of the indices of $\vec{v}$ at $x$ and $- x$ .

The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA

Joseph Cima, Raymond Mortini (1995)

Studia Mathematica

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It is shown that the Bourgain algebra $A {()}_{b}$ of the disk algebra A() with respect to $H^{\infty} ()$ is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to $H^{\infty} ()$ , the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is $A {()}_{b}$ .

Springer fiber components in the two columns case for types $A$ and $D$ are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France

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We study the singularities of the irreducible components of the Springer fiber over a nilpotent element $N$ with $N^{2} = 0$ in a Lie algebra of type $A$ or $D$ (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

Quantum Singularity Theory for $A_{(r - 1)}$ and $r$ -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

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We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$ -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$ -curves is canonically isomorphic to the stack of $r$ -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$ -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine...

Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric Lombardi, Laurent Stolovitch (2010)

Annales scientifiques de l'École Normale Supérieure

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In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of $ℂ^{n}$ , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We study the normal form problem relatively to $S$ . We give a condition on $S$ that ensures...

Real deformations and invariants of map-germs

J. H. Rieger, M. A. S. Ruas, R. Wik Atique (2008)

Banach Center Publications

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A stable deformation $f^{t}$ of a real map-germ $f : ℝ ⁿ, 0 \to ℝ^{p}, 0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^{t}$ are real. A related notion is that of a good real perturbation $f^{t}$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^{t}$ coincides with that of $f_{C}^{t}$ . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good...

An $a b c d$ theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum $1 + u + v$ for two rational functions $u, v$ , in term of the degree of $u, v$ , which is sharp whenever $u, v$ have few distinct zeros and poles compared to their degree. This sharpens the “ $a b c d$ -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^{a} + y^{a} + z^{c} = 1$ contains only finitely many rational or elliptic...

Irregularity of an analogue of the Gauss-Manin systems

Céline Roucairol (2006)

Bulletin de la Société Mathématique de France

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In $𝒟$ -modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+} (𝒪 e^{g})$ of a $𝒟$ -module twisted by the exponential of a polynomial $g$ by another polynomial $f$ , where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

The natural operators $T_{| ℳ f ₙ} ⇝ T * T^{r }$ and $T_{| ℳ f ₙ} ⇝ Λ ² T T^{r *}$

W. M. Mikulski (2002)

Colloquium Mathematicae

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Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{| ℳ f ₙ} ⇝ T * T^{r *}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r *} M = J^{r} (M, ℝ) ₀$ are classified. For n ≥ 3 all natural operators $T_{| ℳ f ₙ} ⇝ Λ ² T * T^{r *}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r *} M$ are completely described.

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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Linear natural operators lifting $p$ -vectors to tensors of type $(q, 0)$ on Weil bundles

Jacek Dębecki (2016)

Czechoslovak Mathematical Journal

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We give a classification of all linear natural operators transforming $p$ -vectors (i.e., skew-symmetric tensor fields of type $(p, 0)$ ) on $n$ -dimensional manifolds $M$ to tensor fields of type $(q, 0)$ on $T^{A} M$ , where $T^{A}$ is a Weil bundle, under the condition that $p \geq 1$ , $n \geq p$ and $n \geq q$ . The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$ -vectors to tensor fields of type $(q, 0)$ on $T^{A}$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

Limits of log canonical thresholds

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $𝒯_{n}$ denote the set of log canonical thresholds of pairs $(X, Y)$ , with $X$ a nonsingular variety of dimension $n$ , and $Y$ a nonempty closed subscheme of $X$ . Using non-standard methods, we show that every limit of a decreasing sequence in $𝒯_{n}$ lies in $𝒯_{n - 1}$ , proving in this setting a conjecture of Kollár. We also show that $𝒯_{n}$ is closed in $𝐑$ ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

Foliations by curves with curves as singularities

M. Corrêa Jr, A. Fernández-Pérez, G. Nonato Costa, R. Vidal Martins (2014)

Annales de l’institut Fourier

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Let $ℱ$ be a holomorphic one-dimensional foliation on $ℙ^{n}$ such that the components of its singular locus $Σ$ are curves $C_{i}$ and points $p_{j}$ . We determine the number of $p_{j}$ , counted with multiplicities, in terms of invariants of $ℱ$ and $C_{i}$ , assuming that $ℱ$ is special along the $C_{i}$ . Allowing just one nonzero dimensional component on $Σ$ , we also prove results on when the foliation happens to be determined by its singular locus.

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...