Coprime values of polynomials in several variables

with P. Dèbes

to appear in Isreal J. Math

Given two polynomials P(x), Q(x) in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values P(n), Q(n) at integer points n being coprime? We show that the set of all gcd (P(n), Q(n)) is stable under gcd and under lcm. A notable consequence is a result of Schinzel: if in addition P and Q have no fixed prime divisor (i.e., no prime dividing all values P(n), Q(n)), then P and Q assume coprime values at "many" integer points. Conversely we show that if "sufficiently many" integer points yield values that are coprime (or of small gcd) then the original polynomials must be coprime. Another noteworthy consequence of this paper is a version over the ring of integers of Hilbert's irreducibility theorem.


The Hilbert-Schinzel specialization property

with P. Dèbes, J. König and S. Najib

to appear in Crelle's Journal

We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in k+n variables, with coefficients in Z, of positive degree in the last n variables, we show that if they are irreducible over Z and satisfy a necessary "Schinzel condition", then the first k variables can be specialized in a Zariski-dense subset of Z^k in such a way that irreducibility over Z is preserved for the polynomials in the remaining n variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first k variables in Z^k, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a "coprime" version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials assume coprime values. We prove our results over many other rings than Z, e.g. UFDs and Dedekind domains for the last one.

Bilipschitz equivalence of polynomials

Mathematische Nachrichten, 294, 2021

We study a family of polynomials in two variables having moduli up to bilip- schitz equivalence: two distinct polynomials of this family are not bilipschitz equivalent. However any level curve of the first polynomial is bilipschitz equivalent to a level curve of the second.

The Schinzel hypothesis for polynomials

with P. Dèbes and S. Najib

Transaction AMS, 373, 2020

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by a polynomial ring and prove the Schinzel hypothesis for a wide class of them: polynomials in at least one variable over the integers, polynomials in several variables over an arbitrary field, etc. We achieve this goal by developing a version over rings of the Hilbert specialization property. A polynomial Goldbach con- jecture is deduced, along with a result on spectra of rational functions.

Prime and coprime values of polynomials

with P. Dèbes and S. Najib

L'enseignement mathématique, 66, 2020

The Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomials and prime numbers. In the same vein, we investigate the common divisors of values of polynomials and establish a "coprime Schinzel Hypothesis". We deduce a version "modulo an integer" of the original Hypothesis, and as special cases, the Goldbach conjecture and the Twin Primes conjecture, again modulo an integer.

Intermediate links of plane curves

with M. Borodzik

Israel J. Math., 227, 2018

For a smooth complex curve we consider the link Lr obtained by intersecting the curve with a sphere of radius r. We prove that the diagram Dr obtained from Lr by a complex stereographic projection satisfies topological equalties. As a consequence we show that if Dr has no negative Seifert circles and Lr is strongly quasipositive and fibered, then the Yamada–Vogel algorithm applied to Dr yields a quasipositive braid.

Families of polynomials and their specializations

with P. Dèbes and S. Najib

J. Number Theory, 170, 2017

For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results of Grothendieck and of Gao.

The braid group of a necklace

with P. Bellingeri

Math. Z., 283, 2016

We study several geometric and algebraic properties of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the con- figuration space of necklaces is isomorphic to the braid group over an annulus. We then defne an action on the fundamental group of the configuration space of the necklaces to the automorphisms of the free group and characterize these automorphisms among all automorphisms of the free group.

Topology of generic line arrangements

Asian J. of Math., 19, 2015

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial f= prod_i (a_ix+b_iy+c_i), so that A= (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, topologically equivalent. In higher dimension the related result is that within a family of equivalent hyperplane arrangements the defining polynomials are topologically equivalent.

Waring problem for polynomials in two variables

with Mireille Car

Proc. American Mathematical Society, 141, 2013

We prove that all polynomials in several variables can be decomposed as the sum of k-th powers: P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+ ... + Q_s(x_1,...,x_n)^k, provided that elements of the base field are themselves sum of k-th powers. We also give bounds for the number of terms s and the degree of the Q_i^k. We then improve these bounds in the case of two variables polynomials to get a decomposition P(x,y) = Q_1(x,y)^k+ ... + Q_s(x,y)^k with deg Q_i^k less than deg P + k^3 and s that depends on k and ln(deg P).

Specializations of indecomposable polynomials

with Guillaume Chèze and Pierre Dèbes

Manuscripta Mathematica, 139, 2012

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial P(x) in Z[x] to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t_1,...,t_r,x) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p=0 or p>deg(f), then the one variable specialized polynomial f(t_1+a_1 x,...,t_r+a_r x,x) is indecomposable for all (t_1,..., t_r, a_1,...,a_r) i the closure of k^{2r} off a proper Zariski closed subset.

Integral points on generic fibers

J. London Mathematical Society, 81, 2010

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

Decomposition of polynomials and approximate roots

Proc. American Mathematical Society, 138, 2010

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.

Generating series for irreducible polynomials

Finite Fields and Applications, 16, 2010

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.

Milnor fibrations of meromorphic functions

with José Seade and Anne Pichon
Journal of the London Mathematical Society, 80, 2009

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.

Indecomposable polynomials and their spectrum

with Pierre Dèbes and Salah Najib
Acta Arithmetica, 139, 2009

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?

Irreducibility of hypersurfaces

with Pierre Dèbes and Salah Najib
Communications in Algebra, 37, 2009

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.

Number of irreducible polynomials over finite fields

American Mathematical Monthly, 115, 2008

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

Jump of Milnor numbers

Bull. Braz. Math. Soc., 38, 2007

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.

Reducibility of rational functions

Israel Journal of Mathematics, 164, 2008

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.

Meromorphic functions, bifurcation sets and fibred links

with Anne Pichon
Mathematical Research Letters, 14 ,2007

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.

Topological equivalence of polynomials

with Mihai Tibar
Advances in Mathematics, 199, 2006

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.

Computation of Milnor numbers and critical values at infinity

Journal of Symbolic Computation, 38, 2004

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.

Newton polygons and families of polynomials

Manuscripta Mathematica, 113, 2004

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).

Irregular fibers of complex polynomials

Revista Matemàtica Complutense, 17, 2004

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.

Invariance of Milnor numbers and topology of polynomials

Commentarii Mathematici Helvetici, 78, 2003

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.

Non reality and non connectivity of polynomials

C. R. Acad. Sci. Paris, 335, 2002

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?

Classification of polynomials with one critical value

Mathematische Zeitschrift, 242, 2002

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

Milnor fibration and fibered links at infinity

International Mathematics Research Notices (IMRN), 11, 1999

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.


The relative Schinzel hypothesis

with P. Dèbes and S. Najib

preprint, 2020

This paper is superseded by the new one: "The Hilbert-Schinzel specialization property", with J. König.

Quelques contributions à la topologie et à l'arithmétique des polynômes

Habilitation Thesis, University Lille 1, July 2008

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...

Fibres et entrelacs irréguliers à l'infini

Ph.D. Thesis, University Toulouse 3, December 2000

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...

Realization of intermediate links of line arrangements

preprint that will not be submitted

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangements; (b) if we intersect the arrangements with a vertical band instead of a sphere, what link configurations can be obtained? (c) relations between link configurations obtained by bands and spheres.

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