Proper trajectories of type {\mathbb{C}^{\ast}} of a polynomial vector field on {\mathbb{C}^2}

We prove that if a polynomial vector field on ${\mathbb{C}^2}$ has a proper and non-algebraic trajectory analytically isomorphic to ${\mathbb{C}^{\ast}}$ all its trajectories are proper, and except at most one which is contained in an algebraic curve of type ${\mathbb{C}}$ all of them are of type ${\mathbb{C}^{\ast}}$ . As corollary we obtain an analytic version of Lin?CZa?denberg Theorem for polynomial foliations.     

On the resolution of singularities in affine toric 3- varieties

Let $x_{\Sigma(\sigma)}=\ {\rm spec C[\check \sigma \cap Z}^{n}]$ be an affine toric variety given by the monoid algebra $\rm C[\check \sigma \cap Z^{n}]$ , $\check \sigma$ the negative dual cone of a lattice cone σ ? Rⁿ, Σ(σ) the fan consisting of the faces of σ. Assume XΣ(σ) to have only quotient singularities. For n = 3 we classify all pairs X_Σ′, X_Σ(σ) which occur in minimal models of equivariant resolutions Φ: X_Σ′ → - X_Σ(σ) sucn that the regular toric variety XΣ′ has Picard number at most 3.     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Center cyclicity of a family of quartic polynomial differential system

In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as

$$\dot{z} = i z + z \bar{z}\big(A z^2 + B z \bar{z} + C \bar{z}^2 \big),$$

where \({A,B,C \in \mathbb{C}}\). We give an upper bound for the cyclicity of any nonlinear center at the origin when we perturb it inside this family. Moreover we prove that this upper bound is sharp.

     

A generalization of Eisenstein�CSch?nemann irreducibility criterion

One of the results generalizing Eisenstein Irreducibility Criterion states that if ${\phi(x) = a_nx^n\,{+} \,a_{n-1}x^{n-1} \,{+} \,\cdots\,{+} \,a_0}$ is a polynomial with coefficients from the ring of integers such that a _s is not divisible by a prime p for some ${s \, \leqslant \, n}$ , each a _i is divisible by p for ${0 \, \leqslant \, i \, \leqslant \, s-1}$ and a ₀ is not divisible by p ², then ${\phi(x)}$ has an irreducible factor of degree at least s over the field of rational numbers. We have observed that if ${\phi(x)}$ is as above, then it has an irreducible factor g(x) of degree s over the ring of p-adic integers such that g(x) is an Eisenstein polynomial with respect to p. In this paper, we prove an analogue of the above result for a wider class of polynomials which will extend the classical Sch?nemann Irreducibility Criterion as well as Generalized Sch?nemann Irreducibility Criterion and yields irreducibility criteria by Akira et?al. (J Number Theory 25:107?C111, 1987).     

Steady States For A Fokker–planck Equation On Sn

We make some considerations over the Fokker-Planck equation ?Δu ? div (uX) = 0, associated to a vector field X on the sphere S _n , obtaining its steady state in cases non Fokker-Planck integrable, proving also that if X is a polynomial vector field, then the sequences {f _i } developing the solution of this equation like $u_\varepsilon = 1 + \sum\nolimits_{i = 1}^\infty {\frac{{f_i }} {{\varepsilon ^i }}}$ are also polynomials.     

Nonsingular star flows satisfy Axiom A and the no-cycle condition

We give an affirmative answer to a problem of Liao and Mañé which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C ¹ neighborhood $\mathcal{U}We give an affirmative answer to a problem of Liao and Ma?é which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C¹ neighborhood in the set of C¹ vector fields such that every singularity and every periodic orbit of every is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition. Dedicated to Shaotao Liao and Ricardo Ma?é Mathematics Subject Classification (2000) 37D30     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

On the k-free values of the polynomial {xy^k + C}

Consider the polynomial \({f(x, y) = xy^k + C}\) for \({k \geq 2}\) and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of \({f(x, y)}\) when \({x, y \leq H}\). We also prove a similar result for the k-free values of \({f(p, q)}\) when \({p, q \leq H}\) are primes, thus extending Erd?s’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.     

The {\bar{\partial}} -Neumann problem on product domains in {\mathbb{C}^{n}}

Let ${\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}$ , where ${\Omega_{j}\subset\mathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${\overline\partial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${\overline\partial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${\overline\partial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.     

Singularities of ball quotients

We prove a result on the singularities of ball quotients ${\Gamma\backslash\mathbb{C}{H^n}}$ by an arithmetic group. More precisely, we show that a ball quotient has at most canonical singularities under certain restrictions on the dimension n and the underlying lattice. We also extend this result to the toroidal compactification.     

A Geometric Analysis of Phase Retrieval

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, \(y_k = \left| \varvec{a}_k^* \varvec{x} \right| \) for \(k = 1, \ldots , m\), is it possible to recover \(\varvec{x} \in \mathbb C^n\) (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors \(\varvec{a}_k\)’s are generic (i.i.d. complex Gaussian) and numerous enough (\(m \ge C n \log ^3 n\)), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal \(\varvec{x}\), up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

The equidistribution of small points for strongly regular pairs of polynomial maps

In this paper, we prove the equidistribution of periodic points of a regular polynomial automorphism $f : \mathbb{A }^n \rightarrow \mathbb{A }^n$ defined over a number field $K$ : let $f$ be a regular polynomial automorphism defined over a number field $K$ and let $v\in M_K$ . Then there exists an $f$ -invariant probability measure $\mu _{f,v}$ on $\mathrm{Berk }\bigl ( \mathbb{P }^n_\mathbb{C _v} \bigr )$ such that the set of periodic points of $f$ is equidistributed with respect to $\mu _{f,v}$ .     

On the entire self-shrinking solutions to Lagrangian mean curvature flow

The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|² D ² u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? _i are the eigenvalues of the Hessian D ² u.     

An algebraic construction of the generic singularities of Boardman-Thom

In this paper we study some of the functorial properties of the infinite jet space in order to give a coordinate free algebraic definition of the generic singularities of Boardman-Thom. More precisely, suppose thatk is a commutative ring with an identity and suppose that A is a commutative ring with an identity which is ak-algebra. An A-k-Lie algebra L is ak-Lie algebra with ak-Lie algebra map ? from L to the algebra ofk-derivations of A to itself such that ford, d′εL anda, a′εA, then $$[ad'],a 'd'] = a(\varphi (d')ad' - a'(\varphi (d')a)d' + aa'[d',\;d'].$$ . There is a universal enveloping algebra for such Lie algebras which we denote by E(L). Denote byL-alg the category of A-algebras B which have L and hence E(L) acting as left operators such that foraεA,dεL, (da)i _B=d(a.i _B). If F is the forgetful functor fromL-alg to the category of A-algebras, we show that F has a left adjoint J(L, ·) which is the natural algebraic translation of the infinite jet space. In the third section of this paper we construct a theory of singularities for a derivation from a ring to a module and then we apply this construction to J(L, C) where C is an A-algebra. These singularities are subschemas with defining sheaf of ideals given by Fitting invariants of appropriately chosen modules when A and B are polynomial rings over a fieldk and C=A? _k B; these are the generic singularities of Boardman-Thom. Finally we show that, under some rather general conditions on the structure of C as an A-algebra, the generic singularities are regular immersions in the sense of Berthelot.     

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution \({\mu _ \in } \in C(\bar \Omega )\) of the Dirichlet problem

$$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$

where Ω is a bounded, open set and \({f_ \in } \in C(\bar \Omega )\) for all ? ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator \(\mathcal{I}_{\in}\) is explicitly given by

$${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$

which is an approximation of the well-known fractional Laplacian of order σ, as ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.

     

Nearly optimal meshes in subanalytic sets

We prove that any fat, subanalytic compact subset of $\mathbb R^N$ possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389?C398, 2010).) We also show that an infinitely differentiable map f from a compact set Q in $\mathbb R^N$ onto a Markov compact set K in $\mathbb C^l$ (l????N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389?C398, 2010) for analytic maps in case Q is a subset of $\mathbb R^N$ . Versions for $\mathcal C^k$ maps with sufficiently large k are also given.