Gilles L a c h a u d 1 S u m m a r y I n t r o d u c t i o n 1. The equation Tq-T = a 2. The equation yq-y = f 3. The genus of coverings 4. Exponential sums and L functions 5. Bounds for traces of exponential sums, for n u m b e r of points of coverings, and for trace equations 6. Examples : coverings of the line 7. The Carlitz-Uchiyama bound for geometric BCH codes Bibliography Introduction

We describe a criterion for showing that the equation s2+y2p = α3 has no non-trivial proper integer solutions for specific primes p > 7. This equation is a special case of the generalized Fermat equation xp + yq + zr = 0. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific p in the final stage...

Hadronic decays of the w’ are reviewed and a new preliminary upper limit of B(~‘+pn )<7.0~10-~ at 90% C.L. from the Mark III is presented. T he I,V’ is the 23S, cc quark-antiquark vector meson and is the only well established radial excitation. The hadronic decay rate of the w’ relative to the J/v should scale as the . ratio of the three gluon widths which are proportional to the leptonic width...

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form yq + y = xm, q being a prime power and m a positive integer which divides q+ 1. The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual c...

In this paper, the partial derivatives Bp,q(x, y) = ∂ ∂xp∂yq B(x, y) of the Beta function B(x, y) are expressed in terms of a finite number of the Polygamma function, where p and q are non-negative integers, x and y are complex numbers. In particular, Bp,q(x, y) can be expressed by the Riemann zeta function if x is equal to n or n + 1 2 and y is equal to m or m + 1 2 , where n and m are integer...

This formula is not satisfactory under a computing point of view because it introduces more algebraic quantities than necessary. The number P(a)/Q'(a) is called the residue of the root a of Q. We introduce the notion of multiplicity of a residue b that is the number of roots a having b as a residue. Trager (1976) introduced a new formula involving less algebraic numbers: let S(y) be the resulta...

The interaction of mercury vacancies with dopant indium atoms in Hg0,7&d0.t,Te was studied using the perturbed ‘y”/ angular correlation (PAC) technique+ Two dominant PAC signals, characterized by quadrupole interaction strengths YQ~ = 83 MHz and yoL = 91 MHz and asymmetry parameters v1 = r/2 = 0.08, were observed and attributed to one or more In-Vu, complexes. The complexes appeared after annea...

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial X p + XYq(p ≥ 2, q > 1) with some admissible symmetry group G defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial X pY + Yq. We will construct an concrete isomorphism between them. The construction is a little bit different in case (p − 1, q) = 1 and case (p...

We describe a criterion for showing that the equation s2+y2p = α3 has no non-trivial proper integer solutions for specific primes p > 7. This equation is a special case of the generalized Fermat equation xp + yq + zr = 0. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific p in the final stage...