Arnaud Bodin - Bibliography

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. See the examples of computations below.

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. See the examples of computations below.

Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to the polynomial x or to x^2-dy^2, d in N. Moreover for such curves (and others) we give a sharp bound for the number

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.

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?

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.

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

In this note we study a problem of A'Campo about the minimal non-zero difference between the Milnor numbers of a germ of plane curve and one of its deformation.

We prove a analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.

We give a necessary condition for a meromorphic function in several variables to give rise to a Milnor fibration of the local link (respectively of the link at infinity). In the case of two variables we give some necessary and sufficient conditions for the local link (respectively the link at infinity) to be fibred.

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

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topology of the polynomials can change. Implementation and examples are given with the computer algebra system Singular.

We consider a continuous family (f_s), s in [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 f_s is constant (up to an algebraic automorphism of C^2).

For a complex polynomial in two variables we study the morphism induced in homology by the embedding of an irregular fiber in a regular neighborhood of it. We give necessary and sufficient conditions for this morphism to be injective, surjective. Particularly this morphism is an isomorphism if and only if the corresponding irregular value is regular at infinity. We apply these results to the study of vanishing and invariant cycles.

We give a global version of Lê-Ramanujam $\mu$-constant theorem for polynomials. Let (f_t), t in [0,1], be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number mu(t), the Milnor number at infinity lambda(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the f_t is a constant, then the polynomials f_0 and f_1 are topologically equivalent. For n>3 we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the f_t are all equivalent.

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?

We give the classification, up to homeomorphisms, of reduced complex polynomials with two variables with one critical value.

For a polynomial f in two complex variables, we prove that the multi-link at infinity of the 0-fiber f^{-1}(0) is a fibred multi-link if and only if all the values different from 0 are regular at infinity.

Les polynômes de plusieurs variables sont présents sous de nombreuses formes en géométrie. L'exemple qui vient immédiatemment à l'esprit est celui d'une courbe algébrique, définie comme le lieu des points qui annulent un polynôme...

Let f from C^2 to C$ be a polynomial map. There exists a set B of irregular values such that f is a locally trivial fibration above C\B. Moreover we know that B = B_aff \cup B_inf where B_inf is the set of irregular values at infinity. In this thesis we study singularities and particularly singularities at infinity...