Showing papers on "Generic polynomial published in 2023"
TL;DR: In this paper , the authors considered polynomial composites with the coefficients from a subset of L and showed that any finite group is a Galois group of some field extensions and solved the inverse Galois problem.
Abstract: In this paper, we consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. We present the characterization of some known field extensions in terms of polynomial composites. This paper contains the open problem of characterization of ideals in polynomial composites with respect to various field extensions. We also present the full possible characterization of certain field extensions. Moreover, in this paper we show that any finite group is a Galois group of some field extensions and present the inverse Galois problem solved.
TL;DR: In this article , the authors considered the Galois group of the Heisenberg Hamiltonian in the XXX integrable model, corresponding to the generic star [k = ± 1, ± 3] of quasimomentum k for octagonal (N = 8) magnetic ring in the two-magnon sector.
08 Mar 2023
TL;DR: Andrews and Petsche as mentioned in this paper showed that the dynamical Galois group of a polynomial with periodic critical orbit is abelian over any quadratic number field.
Abstract: Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,\alpha)$, where $f$ is a polynomial over a number field $K$ and $\alpha\in K$, such that the dynamical Galois group of the pair $(f,\alpha)$ is abelian. In this paper we focus on the case of unicritical polynomials $f$, and more general dynamical systems attached to sequences of unicritical polynomials. After having reduced the conjecture to the post-critically finite case, we establish it for all polynomials with periodic critical orbit, over any number field. We next establish the conjecture in full for all monic unicritical polynomials over any quadratic number field. Finally we show that for any given degree $d$ there exists a finite, explicit set of unicritical polynomials that depends only on $d$, such that if $f=ux^d+1$ is a unicritical polynomial over a number field $K$ that lies outside such exceptional set, then there are at most finitely many basepoints $\alpha$ such that the dynamical Galois group of $(f,\alpha)$ is abelian. To obtain these results, we exploit in multiple ways the group theory of the generic dynamical Galois group to force diophantine relations in dynamical quantities attached to $f$. These relations force in all cases, outside of the ones conjectured by Andrews--Petsche, a contradiction either with lower bounds on the heights in abelian extensions, in the style of Amoroso--Zannier, or with the computation of rational points on explicit curves, carried out with techniques from Balakrishnan--Tuitman and Siksek.
TL;DR: In this paper , the authors give an elementary characterisation of the Galois group of the complex numbers and show that it is a transitive subgroup of a subfield of complex numbers.
Abstract:
Let F be a subfield of the complex numbers and
$f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$
an irreducible polynomial. We give an elementary characterisation of the Galois group of
$f(x)$
as a transitive subgroup of
$S_6$
. The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.