Download Arithmetic, Geometry and Coding Theory (AGCT 2003) by Yves Aubry, Gilles Lachaud PDF

By Yves Aubry, Gilles Lachaud

Résumé :
Arithmétique, géométrie et théorie des codes (AGCT 2003)
En mai 2003 se sont tenus au Centre foreign de Rencontres Mathématiques à Marseille (France), deux événements centrés sur l'Arithmétique, l. a. Géométrie et leurs functions à los angeles théorie des Codes ainsi qu'à los angeles Cryptographie : une école Européenne ``Géométrie Algébrique et Théorie de l'Information'' ainsi que l. a. 9ème édition du colloque overseas ``Arithmétique, Géométrie et Théorie des Codes''. Certains des cours et des conférences font l'objet d'un article publié dans ce quantity. Les thèmes abordés furent à los angeles fois théoriques pour certains et tournés vers des purposes pour d'autres : variétés abéliennes, corps de fonctions et courbes sur les corps finis, groupes de Galois de pro-p-extensions, fonctions zêta de Dedekind de corps de nombres, semi-groupes numériques, nombres de Waring, complexité bilinéaire de los angeles multiplication dans les corps finis et problèmes de nombre de classes.

Mots clefs : Fonctions zêta, variétés abéliennes, corps de fonctions, courbes sur les corps finis, excursions de corps de fonctions, corps finis, graphes, semi-groupes numériques, polynômes sur les corps finis, cryptographie, courbes hyperelliptiques, représentations p-adiques, excursions de corps de classe, groupe de Galois, issues rationels, fractions maintains, régulateurs, nombre de sessions d'idéaux, complexité bilinéaire, jacobienne hyperelliptiques

Abstract:
In may well 2003, occasions were held within the ``Centre foreign de Rencontres Mathématiques'' in Marseille (France), dedicated to mathematics, Geometry and their functions in Coding conception and Cryptography: an eu college ``Algebraic Geometry and data Theory'' and the 9-th overseas convention ``Arithmetic, Geometry and Coding Theory''. a number of the classes and the meetings are released during this quantity. the subjects have been theoretical for a few ones and became in the direction of purposes for others: abelian types, functionality fields and curves over finite fields, Galois team of pro-p-extensions, Dedekind zeta services of quantity fields, numerical semigroups, Waring numbers, bilinear complexity of the multiplication in finite fields and sophistication quantity problems.

Key phrases: Zeta capabilities, abelian types, capabilities fields, curves over finite fields, towers of functionality fields, finite fields, graphs, numerical semigroups, polynomials over finite fields, cryptography, hyperelliptic curves, p-adic representations, classification box towers, Galois teams, rational issues, persisted fractions, regulators, excellent category quantity, bilinear complexity, hyperelliptic jacobians

Class. math. : 14H05, 14G05, 11G20, 20M99, 94B27, 11T06, 11T71, 11R37, 14G10, 14G15, 11R58, 11A55, 11R42, 11Yxx, 12E20, 14H40, 14K05

Table of Contents

* P. Beelen, A. Garcia, and H. Stichtenoth -- On towers of functionality fields over finite fields
* M. Bras-Amorós -- Addition habit of a numerical semigroup
* O. Moreno and F. N. Castro -- at the calculation and estimation of Waring quantity for finite fields
* G. Frey and T. Lange -- Mathematical heritage of Public Key Cryptography
* A. Garcia -- On curves over finite fields
* F. Hajir -- Tame pro-p Galois teams: A survey of modern work
* E. W. Howe, okay. E. Lauter, and J. best -- unnecessary curves of genus 3 and four
* D. Le Brigand -- genuine quadratic extensions of the rational functionality box in attribute two
* S. R. Louboutin -- particular higher bounds for the residues at s=1 of the Dedekind zeta capabilities of a few absolutely actual quantity fields
* S. Ballet and R. Rolland -- at the bilindar complexity of the multiplication in finite fields
* Yu. G. Zarhin -- Homomorphisms of abelian kinds

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Example text

The following result gives a dramatics improvement to Ax-Katz’s, and MorenoMoreno’s results for certain diagonal equations. 2. — Let q = pf and let di be a divisor of q m−1 + q m−2 + · · · + 1 for i = 1, . . , n. Let a1 X1d1 + · · · + an Xndn be a polynomial over Fqml . Then pµ divides |N |, where µ (n − m)lf . ´ ` 11 SEMINAIRES & CONGRES ON WARING NUMBER FOR FINITE FIELDS 31 Let s be the smallest positive integer such that the equation xd1 + · · · + xds = β has at least a solution for every β ∈ Fpf .

2. — Note that cases g(6, 223) > 2, g(8, 761) > 2 imply that the lower bound on pf cannot be improved to (d − 1)3 , since 223 > (6 − 1)3 = 125, 761 > (8 − 1)3 > 716. Let N 4,0 be the number of solutions of the equation xd1 + xd2 + xd3 = βxd4 with x4 = 0 over P(Fpf ) and N 4,1 be the number of solutions of the equation xd1 + xd2 + xd3 = βxd4 with x4 = 0 over P(Fpf ). Now we estimate how large has to be Fpf to obtain that g(d, pf ) 3. 3. — g(d, pf ) 3 whenever p2f > (d−1)(d−2) [2pf /2 ]+ d1 ((d−1)4 +(d−1))pf .

Let f (X, Y ) ∈ Fq [X, Y ] be an absolutely irreducible polynomial such that m := degX f (X, Y ) = degY f (X, Y ). Then we have lim n→∞ #{paths of length n in Γ(f, Fq )} >0 mn if and only if there exists an indecomposable component ∆ of Γ(f, Fq ) whose vertices all have in- and out-degree equal to m. A graph ∆ as in the corollary above has the property that it is a finite indecomposable component of the graph Γ(f, Fq ), since the number of arcs that occur in ∆ is the maximal possible number. Using the above results, we can prove a partial answer to Conjecture 1 (see end of Section 2).

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