Arnaud Bodin

Indecomposable polynomials and their spectrum

k is said to be indecomposable in k[x] if it is not of the form u(H(x)) with H(x)2 k[x] and u2 k[t] with deg(u) 2. An element 2 k is called a spectral value of F (x) if F (x) is reducible in k[x]. It is well-known that (1) F (x)2 k[x] is indecomposable if and only if F (x) is irreducible in k( )[x] (where is an indeterminate), (2) if F (x)2 k[x] is indecomposable, then the subset sp(F ) k of all spectral values of F (x)|the spectrum of F (x)|is nite ; and in the opposite case, sp(F ) = k, (3) more precisely, if F (x) 2 k[x] is indecomposable and for every 2 k, n( ) is the number of irreducible factors of F (x) in k[x], then (F ) := P 2k (n( ) 1) deg(F ) 1. In particular, card(sp(F )) deg(F ) 1. Statement (3), which is known as Stein’s inequality, is due to Stein [13] in characteristic 0 and Lorenzini [10] in arbitrary characteristic (but for two variables); see [11] for the general case. This paper oers some new results in this context. Inx2, given an indecomposable polynomial F (x) with coecients

Number of Irreducible Polynomials in Several Variables over Finite Fields

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

Topological equivalence of complex polynomials

Abstract The following numerical control over the topological equivalence is proved: two complex polynomials in n ≠ 3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions f s : C n → C with isolated singularities such that the degree, the number of vanishing cycles and the number of atypical values are constant in the family.

Newton polygons and families of polynomials

Abstract.We consider a continuous family (fs), s∈[0,1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the fs is constant (up to an algebraic automorphism of ℂ2).

Milnor fibrations of meromorphic functions

In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g: the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that 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.

Decomposition of polynomials and approximate roots

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is the approximate d-root of P. Moreover we give an algorithm to compute such a decomposition. We apply these results to decide whether a polynomial in one or several variables is decomposable or not.

Generating series for irreducible polynomials over finite fields

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials.

Non-reality and non-connectivity of complex polynomials

Abstract Using the same method we provide negative answers to the following questions: is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph)? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomials? To cite this article: A. Bodin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1039–1042.

Waring's problem for polynomials in two variables

We prove that all polynomials in several variables can be decomposed as the sums of