LISTE DES PUBLICATIONS/RÉSUMÉS 1

LISTE DES PUBLICATIONS/RÉSUMÉS
SALAH NAJIB
1. Published or to appear articles
1/ “Indecomposable polynomials via jacobian matrix.”
J. of Algebra (to appear). [With G. Chèze].
Abstract: Indecomposable polynomials are a special class of absolutely irreducible polynomials. Some improvements of important effective results on absolute irreducibility have recently appeared using
Ruppert’s matrix. In a similar way, we show in this paper that the
use of a Jacobian matrix gives sharp bounds for the indecomposability
problem.
2/ “Indecomposable polynomials and their spectrum.”
Acta Arith. 139 (2009), 79-100. [With A. Bodin and P. Dèbes].
Abstract: We address some questions concerning indecomposable
polynomials and their spectrum. How does the spectrum behave via
reduction or specialisation, or via a more general ring morphism? Are
the indecomposability properties equivalent over a field and over its algebraic closure? How many polynomials are decomposable over a finite
field?
3/ “Irreducibility of hypersurfaces.”
Comm. Algebra 37 (2009), no. 6, 1884–1900.
[With A. Bodin and P. Dèbes].
Abstract: Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that
Date: March 4, 2010.
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S. NAJIB
we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P )2 − 1 values of the coefficient. We more generally handle the situation where several specified
coefficients vary.
4/ “Autour d’un théorème de Stein.”
Extracta Math. 23(2) (2008), 173–180.
Abstract: Let K be a field of characteristic zero, K an algebraic
closure of K and P (X, Y ) a non constant polynomial, with coefficients
in K. For λ ∈ K, denote the number of distinct irreducible factors fλ,i
in a factorization of P − λ over K by n(λ). We rewrite without the
“jacobian derivation” aspect of Stein’s proof (1989) forPshowing the
following statement: if P is non composite over K then λ (n(λ) − 1)
is at most equal to deg(P ) − 1.
5/ “Irréductibilité et spécialisation des polynômes.”
Portugaliae Math. 65(3) (2008), 339–343.
Abstract: Starting from the decomposition in base b(X) (a nonconstant polynomial), we give an irreducibility criterion of the bivariate
polynomials. Moreover we study a problem of specialization which has
a link with Hilbert’s irreducibility theorem.
6/ “Un raffinement du caractère hilbertien du corps K(X).”
Manuscripta Math., 120, No. 4 (2006), 415–418.
Abstract: Given an infinite field K. We will show that for the elements of a hilbertian set of K(X) associated to a family of polynomials
with coefficients in K, the specialized polynomials remain also primitive
knowing that the starting polynomials are it. This result refines the
hilbertian character of the field K(X). Moreover, by an application of
this result, one obtains an analog of the famous arithmetic progression
theorem.
7/ “Une généralisation de l’inégalité de Stein-Lorenzini.”
J. of Algebra, 292, Issue 2 (2005), 566–573.
Abstract: Let K be a field of arbitrary characteristic, K an algebraic
closure of K and P a non constant polynomial of n ≥ 2 variables,
with coefficients in K. For λ ∈ K, denote the number of distinct
irreducible factors fλ,i in a factorization of P − λ over K by n(λ).
LISTE DES PUBLICATIONS/RÉSUMÉS
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We show the following statement, which generalizes previous results of
Stein
non composite over K then
P (1989) and Lorenzini (1993): if P isP
λ (n(λ) − 1) is at most equal to minλ {
i deg(fλ,i )} − 1. Moreover,
we give an extension of Theorem 1 of my first paper (see below) for a
positive characteristic.
8/ “Sur le spectre d’un polynôme à plusieurs variables.”
Acta Arithm., 114, No. 2 (2004), 169–181.
Abstract: Given a field K of characteristic 0, an algebraic closure K
of K and a non-constant polynomial P of K[X] = K[X1 , . . . , Xn ], let
σ(P ) be the set of all λ ∈ K such that P (X) − λ is reducible over K.
Q
kλ,i
, where the factors fλ,i
For every λ in σ(P ), write P − λ = n(λ)
i=1 fλ,i
are irreducible
in K[X] and define the numbers ρλ (P ) = n(λ) − 1 and
P
ρ(P ) = λ∈K ρλ (P ). The goal of the paper is to show that for any
set {a1 , . . . , as } of distinct elements of K and for any positive integers
ρ1 , . . . , ρs , there exist a non-composite polynomial P in K[X] such that
σ(P ) = {a1 , . . . , as } and ρai (P ) = ρi for all i = 1, . . . , s. Assuming the
field K is hilbertian, we also show an analog in the case of one variable.
2. PhD Thesis abstarct
“Factorisation des polynômes P (X1, . . . , Xn )−λ et théorème de Stein.”
Université Lille 1, 2005; [advisors: M. Ayad and P. Dèbes].
Abstract : Let K be an arbitrary field of characteristic χ(K) ≥ 0, K
an algebraic closure of K and P (X) := P (X1 , . . . , Xn ) ∈ K[X]\K with
n ≥ 2. For each λ ∈ K, denote the number of distinct irreducible factors
fλ,i of P −λ in K[X] by n(λ). Stein (1989) showed under the hypothesis
K algebraically closed, uncountable with P
χ(K) = 0 and n = 2 the
following result: if P is non composite then λ∈K (n(λ)−1) ≤ deg(P )−
1. Later, for K algebraically closed with χ(K) is arbitrary
and n = 2,
P
Lorenzini
(1993)
improved
this
inequality
to
show:
(n(λ)
− 1) ≤
λ
P
minλ { i deg(fλ,i )} − 1. The aim of this work is twofold. On the one
hand, to extend this better inequality to the case of an arbitrary field
K, with no restriction on χ(K) ≥ 0 and n ≥ 2. On the other hand,
to show that for every set {a1 , . . . , as } of distinct elements of K and
for every positive integers n1 , . . . , ns , there exist a polynomial P (X) in
K[X] which is non composite over K such that n(ai ) = ni , i = 1, . . . , s,
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S. NAJIB
and P − λ is irreducible for each λ ∈
/ {a1 , . . . , as }. Moreover, one can
fix in advance all irreducible factors of the P − λ except one.
Résumé : Soient K un corps quelconque de caractéristique χ(K) ≥ 0,
K une clôture algébrique de K et P (X) := P (X1 , . . . , Xn ) ∈ K[X] \ K,
avec n ≥ 2. Pour tout λ ∈ K, on note n(λ) le nombre de facteurs
irréductibles distincts fλ,i de P − λ dans K[X]. Stein (1989) a montré
sous les hypothèses: K algébriquement clos, non dénombrable avec
χ(K) P
= 0 et n = 2 le résultat suivant: si P est non composé sur K
alors λ∈K (n(λ) − 1) ≤ deg(P ) − 1. Ensuite, pour K algébriquement
clos avec χ(K) quelconque et n = 2, Lorenzini (1993) a amélioré cette
inégalité pour obtenir la forme suivante:
X
X
(n(λ) − 1) ≤ min{
deg(fλ,i )} − 1.
λ
λ
i
L’objet de ce travail est double. D’une part, étendre cette meilleure
inégalité au cas d’un corps K arbitraire, avec χ(K) et n ≥ 2 quelconques. D’autre part, montrer que pour tout ensemble {a1 , . . . , as }
d’éléments distincts de K et pour tous entiers n1 , . . . , ns positifs non
nuls, il existe un polynôme P (X) dans K[X] non composé sur K
tel que n(ai ) = ni , i = 1, . . . , s, et P − λ soit irréductible pour
tout λ ∈
/ {a1 , . . . , as }. De plus, on peut fixer à l’avance les facteurs
irréductibles des P − λ sauf un.
Université de Limoges, IUT du Limousin Site de Brive, Département
GEA. 7 rue Jules Vallès, 19100 Brive. France
E-mail address: [email protected] -- [email protected]