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.     

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.     

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.     

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.     

Strong reducibility of partial numberings

A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershovs completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices.This research has partially been supported by INTAS under grant 00-499 Computability in Hierarchies and Topological Spaces.Mathematics Subject Classification (2000): 03D45     

Dual Garside structure and reducibility of braids

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a standard curve to another standard curve, then the image of this curve under the action of each factor of the left normal form of the braid (with the classical Garside structure) is also standard. We provide a new simple, geometric proof of the result by Benardete–Gutierrez–Nitecki, which can be easily adapted to the case of the dual Garside structure of braid groups, with the appropriate definition of standard curves in the dual setting. This yields a new algorithm for determining the Nielsen–Thurston type of braids.