Hilbert Property Research Articles

Hilbert properties under base change in small extensions

Hilbert properties under base change in small extensions

Weak weak approximation and the Hilbert property for degree 2 del Pezzo surfaces

AbstractWe prove that del Pezzo surfaces of degree 2 over a field satisfy weak weak approximation if is a number field and the Hilbert property if is Hilbertian of characteristic zero, provided that they contain a ‐rational point lying neither on any 4 of the 56 exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado–Testa–Várilly‐Alvarado, and Festi–van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration.

Weak Approximation and the Hilbert Property for Campana Points

Weak Approximation and the Hilbert Property for Campana Points

Large rank jumps on elliptic surfaces and the Hilbert property

Large rank jumps on elliptic surfaces and the Hilbert property

A Hilbert irreducibility theorem for Enriques surfaces

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0 0 . Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank ≥ 6 \geq 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

Rank jumps on elliptic surfaces and the Hilbert property

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.

The Hilbert property for arithmetic schemes

We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker–Fehm–Petersen and Corv

Rational points and ramified covers of products of two elliptic curves

Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata’s structure result for ramified covers of abelian varie

Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type

AbstractIn studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

Non-rational varieties with the Hilbert Property

A variety [Formula: see text] over a field [Formula: see text] is said to have the Hilbert Property if [Formula: see text] is not thin. We shall exhibit some examples of varieties, for which the Hilbert Property is a new result. We give a sufficient condition for descending the Hilbert Property to the quotient of a variety by the action of a finite group. Applying this result to linear actions of groups, we exhibit some examples of non-rational unirational varieties with the Hilbert Property, providing positive instances of a conjecture posed by Colliot–Thélèene and Sansuc. We also give a sufficient condition for a surface with two elliptic fibrations to have the Hilbert Property, and use it to prove that a certain class of K3 surfaces have the Hilbert Property, generalizing a result of Corvaja and Zannier.

Elliptic Fibrations and the Hilbert Property

AbstractFor a number field $K$, an algebraic variety $X/K$ is said to have the Hilbert Property if $X(K)$ is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a $K$-rational point in ${\mathbb{P}}_n/K$, for $n \geq 3$. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.

On the Hilbert property and the fundamental group of algebraic varieties

In this paper we link the so-called Hilbert property (HP) for an algebraic variety (over a number field) with its fundamental group, in a perspective which appears new. (The notion of HP comes from Hilbert’s Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem.) We shall observe that the HP is in a sense ‘opposite’ to the Chevalley–Weil Theorem. This shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the HP, involving Enriques surfaces. In this view, we shall also formulate an alternative related property, possibly holding in full generality, and some conjectures which unify part of what is known in the topic. These predict that for a variety with a Zariski-dense set of rational points, the validity of the HP, which is defined arithmetically, is indeed of purely topological nature. Also, a consequence of these conjectures would be a positive solution of the Inverse Galois Problem. In the paper we shall also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a variety non rational (over \({\mathbb {C}}\)) for which the HP can be proved. In an Appendix we shall further discuss, among other things, the HP also for Kummer surfaces. All of this basically exhausts the study of the HP for surfaces with a Zariski-dense set of rational points.

Evolutionary Spectral Analyses of a Powerful Typhoon at the Sutong Bridge Site Based on the HHT

To investigate the nonstationary characteristics of strong typhoons, this paper considers the evolutionary spectral characteristics of strong typhoons based on the Hilbert-Huang transform (HHT). Discrete expressions are determined for the evolutionary spectral analysis based on the HHT. The study indicates that the classic empirical mode decomposition (EMD) method fails to extract all of the high-frequency fluctuations from wind velocity data, and the time-averaged power spectrum obtained directly using the HHT cannot provide the true wind velocity spectrum. The degrees of nonstationarity of different-order IMF components are analysed and a synthesized method of analysing the evolutionary spectrum and time-averaged power spectrum of a strong typhoon is proposed. To avoid the energy leakage problem that exists in HHT spectral analyses, the Gram-Schmidt method is used to orthogonalize the intrinsic mode function (IMF) components. The study indicates that when the orthogonalization is implemented in accordance with the sequence from high-order IMF components to low-order ones, the orthogonalized components retain the same good Hilbert property as that of the IMFs. The synthesized method proposed yields a time-averaged power spectrum that is consistent with the Fourier spectrum in value and can produce the energy distributions of a typhoon in the time and frequency domains simultaneously.

Learning Rates for Regularized Classifiers Using Trigonometric Polynomial Kernels

Regularized classifiers are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the trigonometric polynomial kernels are ones of the most important kernels and play key roles in signal processing. The main target of this paper is to provide convergence rates of classification algorithms generated by regularization schemes with trigonometric polynomial kernels. As a special case, an error analysis for the support vector machines (SVMs) soft margin classifier is presented. The norms of Fejer operator in reproducing kernel Hilbert space and properties of approximation of the operator in L 1 space with periodic function play key roles in the analysis of regularization error. Some new bounds on the learning rate of regularization algorithms based on the measure of covering number for normalized loss functions are established. Together with the analysis of sample error, the explicit learning rates for SVM are also derived.

The Bessel and Hilbert properties for root functions of second-order ordinary differential operators with the Carleman condition violated

The Bessel and Hilbert properties for root functions of second-order ordinary differential operators with the Carleman condition violated

Hilbert and Bessel Properties of Systems of Root Functions of 2mth-Order Operators

Hilbert and Bessel Properties of Systems of Root Functions of 2mth-Order Operators

The hilbert inequality for a one-parameter family of functions

etc. Particular systems of this kind were considered by numerous authors (e.g., see [1-5]), and it always turned out that the set of values of c~ for which the system is a basis (or a conditional basis) in L2(0, 1) is an open set. Since the investigation of the unconditional basis property of a system in L2 can be reduced to the analysis of the Hilbert and Bessel properties and the minimality for a normalized system in L2, we naturally encounter the following problem: what is the set of values of c~ for which the system {u~(x,c~)}~__t has the nilbert and Bessel properties in L2(a,b) [here II" II is the norm in L2(a,b)]? Recall that a system {e~}~__l in a Hilbert space H is called a Hilbert system if there exists a 7 > 0 such that the Hilbert inequality

On Multidimensional Curves with Hilbert Property

Indexing schemes for grids based on space-filling curves (e.g., Hilbert curves) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multidimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling indexing schemes. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability. We define and analyze in a precise mathematical way r -dimensional Hilbert curves for arbitrary r ≥ 2 . Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multidimensional grids. As we show, curves with Hilbert property can be completely described and analyzed by ``generating elements of order 1,'' thus, in comparison with previous work, reducing their structural complexity decisively. Whereas there is basically one Hilbert curve in the two-dimensional world, our analysis shows that there are 1536 structurally different simple three-dimensional Hilbert curves. Further results include generalizations of locality results for multidimensional indexings and an easy recursive computation scheme for multidimensional curves with Hilbert property. In addition, our formalism lays the groundwork for potential mechanized analysis of locality properties of multidimensional Hilbert curves.

The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators

The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators

Note on g -Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂¯Q such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin.