Éditeur(s) : HAL CCSD
Résumé : In this paper we define the $p$-density of a finite subset $Dsubsetma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.
https://hal.archives-ouvertes.fr/hal-00551478

hal-00551478
https://hal.archives-ouvertes.fr/hal-00551478
ARXIV : 0812.3382

Éditeur(s) : HAL CCSD
Résumé : Note, 5 pages
We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M_{(A,E,P)}/N_{(I_{A},E,P)}. With this new definition and in the particular case of E=C^{∞}, we show that a moderate element i.e. in M_{(A,E,P)} is negligible if and only if it satisfies the C⁰-order estimate for the ideal N_{(I_{A},E,P)}.
https://hal.archives-ouvertes.fr/hal-00326671

Droits : info:eu-repo/semantics/OpenAccess
hal-00326671
https://hal.archives-ouvertes.fr/hal-00326671
https://hal.archives-ouvertes.fr/hal-00326671/document
https://hal.archives-ouvertes.fr/hal-00326671/file/AustrianLemmaVer1.pdf
ARXIV : 0810.0831

Éditeur(s) : HAL CCSD
Résumé : 18 pages
This article adresses questions on localization and microlocalization in the framework of the so called (C,E,P)-algebras of generalized functions. We give new results on nonlinear properties of the asymptotic singular spectrum in the multi-parameter case, a tool to describe the singularities of generalized solutions associated to problems which are classically ill-posed. Wa apply the theory to a transport equation with strongly irregular coefficients and initial data.
https://hal.univ-antilles.fr/hal-00761112

hal-00761112
https://hal.univ-antilles.fr/hal-00761112
https://hal.univ-antilles.fr/hal-00761112/document
https://hal.univ-antilles.fr/hal-00761112/file/DT2010-03_Devoue_al3.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We show that a new definition of the asymptotic structure of the (C,E,P)-algebras (which are a refinement of the Colombeau Generalized Algebras) allows to define a natural topology on them. Furthermore, this topology can be mixed with functorial aspects of the construction in order to give extension principles, which give powerful tools to solve singular differential problems in this framework.
ISSN: 0022-247X

hal-01529551
https://hal.univ-antilles.fr/hal-01529551
DOI : 10.1016/j.jmaa.2009.05.046

Éditeur(s) : HAL CCSD
Résumé : We extend classical results from the Colombeau algebra, concerning point-value characterizations of generalized functions, to the more general case of multi-parameter (C,E,P)-algebras. Our investigations include considerations about different definitions of subspaces related to tempered generalized functions.
https://hal.univ-antilles.fr/hal-00761114

hal-00761114
https://hal.univ-antilles.fr/hal-00761114
https://hal.univ-antilles.fr/hal-00761114/document
https://hal.univ-antilles.fr/hal-00761114/file/DT2010-05_Hasler_Marti.pdf

Éditeur(s) : HAL CCSD
Résumé : In this paper we compute the first vertex of a generic Newton polygon in some special cases, and the corresponding Hasse polynomial. This allows us to show the nonexistence of $p$-cyclic coverings of the projective line in characteristic $p$ with supersingular jacobian for some (infinite families of) genera.
https://hal.archives-ouvertes.fr/hal-00551456

hal-00551456
https://hal.archives-ouvertes.fr/hal-00551456
ARXIV : 0912.2051

Éditeur(s) : HAL CCSD Chapman & Hall/CRC
Résumé : International audience
Nonlinear Theory of Generalized Functions

ISBN : 9780849306495
hal-01529543
https://hal.univ-antilles.fr/hal-01529543

Éditeur(s) : HAL CCSD
Résumé : International audience
8th Congress of the International Society for Analysis, its Applications, and Computation
Moscou, Russia

hal-01529545
https://hal.univ-antilles.fr/hal-01529545

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Let C be a smooth curve over a Galois ring R. Let f be a function over C, and Ψ be an additive character of order p^l over R; in this paper we study the exponential sums associated to f and Ψ over C, and their L-functions. We show the rationality of the L-functions in a more general setting, then in the case of curves we express them as products of L-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the L-functions are, up to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums.
ISSN: 1071-5797

hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470/document
https://hal.archives-ouvertes.fr/hal-00551470/file/Charlefinal.pdf
DOI : 10.1016/j.ffa.2009.01.001