Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski; Maria Przybylska

Displaying similar documents to “Overview of the differential Galois integrability conditions for non-homogeneous potentials”

Random Galois extensions of Hilbertian fields

Lior Bary-Soroker, Arno Fehm (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let $L$ be a Galois extension of a countable Hilbertian field $K$ . Although $L$ need not be Hilbertian, we prove that an abundance of large Galois subextensions of $L / K$ are.

Counting discriminants of number fields

Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)

Journal de Théorie des Nombres de Bordeaux

Similarity:

For each transitive permutation group $G$ on $n$ letters with $n \leq 4$ , we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $ℚ$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$ .

Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei (2014)

Publications mathématiques de Besançon

Similarity:

Fix an integer $ℓ \geq 3$ . Rikuna introduced a polynomial $r (x, t)$ defined over a function field $K (t)$ whose Galois group is cyclic of order $ℓ$ , where $K$ satisfies some mild hypotheses. In this paper we define the family of ${r_{n} (x, t)}_{n \geq 1}$ of degree $ℓ^{n}$ . The $r_{n} (x, t)$ are constructed iteratively from the $r (x, t)$ . We compute the Galois groups of the $r_{n} (x, t)$ for odd $ℓ$ over an arbitrary base field and give applications to arithmetic dynamical systems.

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

Similarity:

Let $ε$ be an algebraic unit of the degree $n \geq 3$ . Assume that the extension $ℚ (ε) / ℚ$ is Galois. We would like to determine when the order $ℤ [ε]$ of $ℚ (ε)$ is $Gal (ℚ (ε) / ℚ)$ -invariant, i.e. when the $n$ complex conjugates $ε_{1}, \dots, ε_{n}$ of $ε$ are in $ℤ [ε]$ , which amounts to asking that $ℤ [ε_{1}, \dots, ε_{n}] = ℤ [ε]$ , i.e., that these two orders of $ℚ (ε)$ have the same discriminant. This problem has been solved only for $n = 3$ by using an explicit formula for the discriminant of the order $ℤ [ε_{1}, ε_{2}, ε_{3}]$ . However, there is no known similar formula for $n > 3$ . In the present paper, we put forward and...

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(α)} (x)$ is a non-zero square for some integer pair $(n, α)$ , with $n \geq 1$ , if and only if $(n, α)$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is the alternating group $A_{n}$ . For example, we establish that for all but finitely many positive integers $n \equiv 2 (mod 4)$ , the only $α$ for which the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is $A_{n}$ is...

Preperiodic dynatomic curves for $z \mapsto z^{d} + c$

Yan Gao (2016)

Fundamenta Mathematicae

Similarity:

The preperiodic dynatomic curve $_{n, p}$ is the closure in ℂ² of the set of (c,z) such that z is a preperiodic point of the polynomial $z \mapsto z^{d} + c$ with preperiod n and period p (n,p ≥ 1). We prove that each $_{n, p}$ has exactly d-1 irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of $_{n, p}$ . We also compute the genus of each component and the Galois group of the defining polynomial of $_{n, p}$ .

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of...

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Matilde Marcolli, Gonçalo Tabuada (2016)

Journal of the European Mathematical Society

Similarity:

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum ${(k)}_{F}$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum ${(k)}_{F}$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined...

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

Similarity:

Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

Piecewise hereditary algebras under field extensions

Jie Li (2021)

Czechoslovak Mathematical Journal

Similarity:

Let $A$ be a finite-dimensional $k$ -algebra and $K / k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A \otimes_{k} K$ .