On the reducibility of some composite polynomials over finite fields

Let g(x)?=?x ⁿ ?+?a _n-1 x ^n-1?+?. . .?+?a ₀ be an irreducible polynomial over ${\mathbb{F}_q}$ . Varshamov proved that for a?=?1 the composite polynomial g(x ^p ?ax?b) is irreducible over ${\mathbb{F}_q}$ if and only if ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a?=?1 and ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$ and for the case a?≠ 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form ${g(x^{r^kp}-x^{r^k})}$ are also presented. Moreover, Cohen’s method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q ³ ? q)?=?1.     

Generic polynomial vector fields are not integrable

We study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter.Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.     

Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems

In this paper, we prove that a quasi-periodic linear differential equation in sl(2,?) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number and give some applications for Schrödinger operators. Our proof is a generalized KAM type iteration adapted to all irrational α.     

有限域上与仿射多项式有关的一些多项式的可约性

在这篇文章中,研究了有限域上一些与仿射多项式有关的多项式的可约性.对于有限域Fp上不是xppt-x-1的仿射三项式,得到了这些三项式的一个明确的因式.完全确定了多项式g(xps-ax-b)在Fp[x]中的分解,这里g(x)是Fp[x]中一个不可约多项式.证明了Fp上次数相同的不可约多项式的全体可以构成一个正则图.同时给出了多项式g(xqs-x-b)在Fp[x]不可约因式的个数公式,这里g(x)是Fp上一个不可约多项式.     

Some presentations for the special linear groups on finite fields

We simplify the Steinberg presentation of SL _n (F _d ), where n≥1 and F _d is any finite field with d elements. That presentation has the elementary matrices e _ij (r), with i,j∈{1,...,n}, i≠=j and r∈F _d , as generators, and (E1)–(E3), described at the opening of this work, as relations. The presentation that we shall obtain reduces the number of generators e _ij (r) and relations (E1)–(E3). In particular, relations (E3) are considerably reduced. Received: March 16, 1998; in final form: November 3, 2000?Published online: October 2, 2001     

Generic super-exponential stability of elliptic equilibrium positions for symplectic vector fields

In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. We will focus on the neighborhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in a Hamiltonian system should be similar. More specifically, Morbidelli and Giorgilli have proved a result of stability over superexponentially long times if one considers an analytic Lagrangian torus, invariant for an analytic Hamiltonian system, with a diophantine translation vector which admits a sign-definite torsion. Then, the solutions of the system move very little over times which are super-exponentially long with respect to the inverse of the distance to the invariant torus. The proof proceeds in two steps: first one constructs a high-order Birkhoff normal form, then one applies the Nekhoroshev theory. Bounemoura has shown that the second step of this construction remains valid if the Birkhoff normal form linked to the invariant torus or the elliptic fixed point belongs to a generic set among the formal series. This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that the most strongly non resonant elliptic fixed point or invariant torus in a Hamiltonian system admits Birkhoff normal forms fitted for the application of the Nekhoroshev theory. Actually, the set introduced by Bounemoura is already very large but not big enough to ensure that a typical Birkhoff normal form falls into this class. We show here that this property is satisfied generically in the sense of the measure (prevalence) through infinite-dimensional probe spaces (that is, an infinite number of parameters chosen at random) with methods similar to those developed in a paper of Gorodetski, Kaloshin and Hunt in another setting.     

Nielsen-Thurston reducibility and renormalization

Consider an orientation preserving homeomorphism of the 2-disk with an infinite set of nested periodic orbits , such that, for all , the restriction of to the complement of the first orbits, from to , is times reducible in the sense of Nielsen and Thurston. We define combinatorial renormalization operators for such maps, and study the fixed points of these operators. We also recall the corresponding theory for endomorphisms of the interval, and give elements of comparison of the theories in one and two dimensions.

     

Flow hypergraph reducibility

Reducible flowgraphs were first defined by Allen in terms of intervals; another definition based on two flowgraph transformations was presented by Hecht and Ullman. In this paper, we extend the notion of reducibility to directed hypergraphs, proving that the interval and the transformation approaches preserve the equivalence when applied to this family.     

On presentations of integer polynomial points of simple groups over number fields

Let K be a number field and let A be its ring of integers. Let G be a connected, noncommutative, absolutely almost simple algebraic K-group. If the K-rank of G equals 2, then G(A[t]) is not finitely presented.     

Complete reducibility and Steinberg endomorphisms

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our main theorem extends a special case of a rationality result in this setting.