Related papers: Th\'eor\`eme de Chebotarev effectif

We investigate the number of prime factors of individual entries for matrices in the special linear group over the integers. We show that, when properly normalised, it satisfies a central limit theorem of Erd\H{o}s-Kac-type. To do so, we…

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this…

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of…

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion…

Given a number field $K$ that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers $\mathfrak{O}_K$ of $K$ by describing certain Frobenius semigroups,…

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all…

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual,…

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of…

It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal…

For a non-CM elliptic curve $E$ defined over the rationals, Lang and Trotter made very deep conjectures concerning the number of primes $p\leq x$ for which $a_p(E)$ is a fixed integer (and for which the Frobenius field at $p$ is a fixed…

Let $X$ be an Abelian group of the form $X=\mathbb{R}^m\times K\times D$, where $m\geq 0$, $K$ is a compact totally disconnected group of the special form, $D$ is a discrete group. Let $\xi_i, i=1,2,...,n,n\geq 2,$ be independent random…

Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group…

In this note we consider the links of prime ideals of certain skew polynomial rings and prove our main theorem, namely theorem [5], which states the following.Let R be a noetherian ring that is link k-symmetric and let {\sigma} be an…

We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group ${\bf G}(\mathbb{A}_{\mathbb{K}})$, associated to the data of a `Whittaker pair'. We describe a quasi-order on Fourier…

There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or…

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random…

When $G$ is solvable group, we prove that the number of conjugacy classes of elements of prime power order is less than or equal to the number of irreducible characters with values in fields where $\mathbb {Q}$ is extended by prime power…

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on…

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials.…

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and…