Transactions of the American Mathematical Society, 1995
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
Proceedings of the American Mathematical Society, Jan 29, 2014
Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk ... more Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ − a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture.
This paper is concerned with the study of Mahler's measure of an univariate polynomial. A theorem... more This paper is concerned with the study of Mahler's measure of an univariate polynomial. A theorem of Szego says that the measure of P is equal to the infimum of &PQ& 2 where Q is a monic polynomial. Here we study how the infimum of &PQ& 2 , where Q is monic and has degree k, tends to the measure of P when k tends to infinity: this answers a question raised by Cerlienco, Mignotte and Piras [
Le calcul de la mesure de Mahler d'un polynome a coefficients complexes en une variable a par... more Le calcul de la mesure de Mahler d'un polynome a coefficients complexes en une variable a partir de ses coefficients est un probleme difficile. Pour le resoudre, on utilise souvent les polynomes iteres de Graeffe. Ici, nous construisons deux suites, a partir de ces polynomes, qui convergent vers cette mesure. Ces suites sont respectivement fonctions de la norme L 2 et de la norme de Bombieri des polynomes iteres et sont facilement implementables sur machine. On compare leurs vitesses de convergence et on remarque qu'elles dependent de la localisation des racines du polynome
Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des ... more Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des polynomes en une ou plusieurs variables, avec une preoccupation commune: disposer d'outils quantitatifs effectifs, permettant des calculs explicites. Le premier chapitre (realise en collaboration avec b. Beauzamy et a paraitre aux transactions a. M. S. ) permet d'etablir une identite differentielle qui ameliore une inegalite de bombieri. Le second chapitre represente une premiere investigation de la mesure de mahler d'un polynome en une variable: rapidite de la convergence des iteres de graeffe. Il a ete realise en collaboration avec j. C. Hohl et odile jenvrin, et fait l'objet d'une note de comptes rendus de l'academie des sciences. Le troisieme chapitre est consacre a la construction massivement parallele des polynomes d'interpolation en nombreuses variables et a l'etude de la stabilite de l'algorithme. Le quatrieme chapitre concerne le calcul de la ...
Proceedings of the American Mathematical Society, Jun 21, 2011
Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the clos... more Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture.
Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des ... more Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des polynomes en une ou plusieurs variables, avec une preoccupation commune: disposer d'outils quantitatifs effectifs, permettant des calculs explicites. Le premier chapitre (realise en collaboration avec b. Beauzamy et a paraitre aux transactions a. M. S. ) permet d'etablir une identite differentielle qui ameliore une inegalite de bombieri. Le second chapitre represente une premiere investigation de la mesure de mahler d'un polynome en une variable: rapidite de la convergence des iteres de graeffe. Il a ete realise en collaboration avec j. C. Hohl et odile jenvrin, et fait l'objet d'une note de comptes rendus de l'academie des sciences. Le troisieme chapitre est consacre a la construction massivement parallele des polynomes d'interpolation en nombreuses variables et a l'etude de la stabilite de l'algorithme. Le quatrieme chapitre concerne le calcul de la mesure de mahler en dimension finie, et repond a une question posee par cierlienco-mignotte-piras en 1987. Publications: (1) beauzamy, b - degot, j: differential identities. Accepte pour publication aux transactions a. M. S. . (2) degot, j - hohl j. C. - jenvrin, o: calcul numerique de la mesure de mahler d'un polynome par iterations de graeffe. Accepte pour publication aux notes de comptes rendus, acad. Sci.
Sendov conjecture says that any complex polynomial P having all its zeros in the closed unit disk... more Sendov conjecture says that any complex polynomial P having all its zeros in the closed unit disk and a being one of them, the closed disk of center a and radius 1 contains a zero of the derivative P ′. The main result of this paper is a proof of Sendov conjecture when the polynomial P has a degree higher than a fixed integer N. We will give estimates of this integer N with respect to |a|. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of P′) of a polynomial which will eventualy contradict Sendov conjecture.
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
Abstract In this paper, we present homogeneous polynomials in many variables. We show how the hyp... more Abstract In this paper, we present homogeneous polynomials in many variables. We show how the hypercube representation of these polynomials (introduced by Beauzamy et al. in [1], and derived from Bombieri's work in Beauzamy et al. [2]) allows us to build interpolation polynomials, that is, polynomials taking prescribed values at prescribed points in C N . We then show that the construction is robust and give quantitative estimates on how the constructed polynomial is perturbed if either the data, the points, or both are perturbed. The theorems, constructions, and algorithms answer questions asked by Dr. Ken Clark, U.S. Army Research Office. In the final part of the paper, we present the explicit algorithms, implemented on the Connection Machines CM200 and CM5 at the Etablissement Technique Central de l'Armement, Arcueil. This algorithm is efficient, especially when the number of variables is high, and it takes all advantage of the massively parallel architecture.
Transactions of the American Mathematical Society, 1995
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements.
Transactions of the American Mathematical Society, 1995
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
Proceedings of the American Mathematical Society, Jan 29, 2014
Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk ... more Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ − a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture.
This paper is concerned with the study of Mahler's measure of an univariate polynomial. A theorem... more This paper is concerned with the study of Mahler's measure of an univariate polynomial. A theorem of Szego says that the measure of P is equal to the infimum of &PQ& 2 where Q is a monic polynomial. Here we study how the infimum of &PQ& 2 , where Q is monic and has degree k, tends to the measure of P when k tends to infinity: this answers a question raised by Cerlienco, Mignotte and Piras [
Le calcul de la mesure de Mahler d'un polynome a coefficients complexes en une variable a par... more Le calcul de la mesure de Mahler d'un polynome a coefficients complexes en une variable a partir de ses coefficients est un probleme difficile. Pour le resoudre, on utilise souvent les polynomes iteres de Graeffe. Ici, nous construisons deux suites, a partir de ces polynomes, qui convergent vers cette mesure. Ces suites sont respectivement fonctions de la norme L 2 et de la norme de Bombieri des polynomes iteres et sont facilement implementables sur machine. On compare leurs vitesses de convergence et on remarque qu'elles dependent de la localisation des racines du polynome
Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des ... more Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des polynomes en une ou plusieurs variables, avec une preoccupation commune: disposer d'outils quantitatifs effectifs, permettant des calculs explicites. Le premier chapitre (realise en collaboration avec b. Beauzamy et a paraitre aux transactions a. M. S. ) permet d'etablir une identite differentielle qui ameliore une inegalite de bombieri. Le second chapitre represente une premiere investigation de la mesure de mahler d'un polynome en une variable: rapidite de la convergence des iteres de graeffe. Il a ete realise en collaboration avec j. C. Hohl et odile jenvrin, et fait l'objet d'une note de comptes rendus de l'academie des sciences. Le troisieme chapitre est consacre a la construction massivement parallele des polynomes d'interpolation en nombreuses variables et a l'etude de la stabilite de l'algorithme. Le quatrieme chapitre concerne le calcul de la ...
Proceedings of the American Mathematical Society, Jun 21, 2011
Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the clos... more Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture.
Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des ... more Ce travail est constitue de quatre articles, consacres a l'etude de certaines proprietes des polynomes en une ou plusieurs variables, avec une preoccupation commune: disposer d'outils quantitatifs effectifs, permettant des calculs explicites. Le premier chapitre (realise en collaboration avec b. Beauzamy et a paraitre aux transactions a. M. S. ) permet d'etablir une identite differentielle qui ameliore une inegalite de bombieri. Le second chapitre represente une premiere investigation de la mesure de mahler d'un polynome en une variable: rapidite de la convergence des iteres de graeffe. Il a ete realise en collaboration avec j. C. Hohl et odile jenvrin, et fait l'objet d'une note de comptes rendus de l'academie des sciences. Le troisieme chapitre est consacre a la construction massivement parallele des polynomes d'interpolation en nombreuses variables et a l'etude de la stabilite de l'algorithme. Le quatrieme chapitre concerne le calcul de la mesure de mahler en dimension finie, et repond a une question posee par cierlienco-mignotte-piras en 1987. Publications: (1) beauzamy, b - degot, j: differential identities. Accepte pour publication aux transactions a. M. S. . (2) degot, j - hohl j. C. - jenvrin, o: calcul numerique de la mesure de mahler d'un polynome par iterations de graeffe. Accepte pour publication aux notes de comptes rendus, acad. Sci.
Sendov conjecture says that any complex polynomial P having all its zeros in the closed unit disk... more Sendov conjecture says that any complex polynomial P having all its zeros in the closed unit disk and a being one of them, the closed disk of center a and radius 1 contains a zero of the derivative P ′. The main result of this paper is a proof of Sendov conjecture when the polynomial P has a degree higher than a fixed integer N. We will give estimates of this integer N with respect to |a|. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of P′) of a polynomial which will eventualy contradict Sendov conjecture.
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
Abstract In this paper, we present homogeneous polynomials in many variables. We show how the hyp... more Abstract In this paper, we present homogeneous polynomials in many variables. We show how the hypercube representation of these polynomials (introduced by Beauzamy et al. in [1], and derived from Bombieri's work in Beauzamy et al. [2]) allows us to build interpolation polynomials, that is, polynomials taking prescribed values at prescribed points in C N . We then show that the construction is robust and give quantitative estimates on how the constructed polynomial is perturbed if either the data, the points, or both are perturbed. The theorems, constructions, and algorithms answer questions asked by Dr. Ken Clark, U.S. Army Research Office. In the final part of the paper, we present the explicit algorithms, implemented on the Connection Machines CM200 and CM5 at the Etablissement Technique Central de l'Armement, Arcueil. This algorithm is efficient, especially when the number of variables is high, and it takes all advantage of the massively parallel architecture.
Transactions of the American Mathematical Society, 1995
We deal here with homogeneous polynomials in many variables and their hypercube representation, i... more We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements.
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