Algebraic Coding: First French-Soviet Workshop Paris, July - download pdf or read online By A. Tietäväinen (auth.), Gérard Cohen, Antoine Lobstein, Gilles Zémor, Simon Litsyn (eds.)

ISBN-10: 3540551301

ISBN-13: 9783540551300

This quantity provides the lawsuits of the 1st French-Soviet workshop on algebraic coding, held in Paris in July 1991. the assumption for the workshop, born in Leningrad (now St. Petersburg) in 1990, used to be to collect the very best Soviet coding theorists. Scientists from France, Finland, Germany, Israel, Italy, Spain, and the us additionally attended. The papers within the quantity fall quite certainly into 4 different types: - functions of exponential sums - protecting radius - buildings -Decoding.

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5. Affine Toric varieties In this section we will show that any normal quasi-ordinary singularity is a toric affine variety. In , P. 2 stated below, for quasi-ordinary 3 hypersurfaces of C . D. 4], P. Popescu-Pampu gave an other proof for the same result, and as he says, his proof extends to general normal quasiordinary singularities. We give here a “hand-made” proof of that theorem. Let Γ be a subgroup of Zn of finite index. 1. If we call v1 , . . , vn the rows of the matrix M −1 , then (det M )vi is the ith row of the adjoint matrix Adj M .

Proof. — Let Γ be the subgroup of a quasi-ordinary projection associated to (X, 0). 1. 2, (X, 0) is isomorphic to the normalization of (XAdj M,det M , 0). 3. — Let Γ be the subgroup of Z2 generated by the lower triangular system {(1, −1), (0, 2)}. Then any normal quasi-ordinary singularity of dimension 2 having Γ as subgroup for some quasi-ordinary projection is isomorphic to the normalization of an irreducible component of the space defined in C4 by: z12 = x21 z22 = x1 x2 . It is then isomorphic to the hypersurface of C3 defined by z 2 = xy.

Choose a system of coordinates (x1 , . . , xn ) in U , in such a way that the branching locus of f is contained in the space H defined by x1 · · · xn = 0. Set U ∗ = U H and X ∗ = X f −1 (H). The restricted map f : X ∗ → U ∗ is a topological covering. The space U ∗ is homeomorphic to the complex torus C∗n . Since π1 (U ∗ ) Zn is abelian, the image of the induced map f∗ : π1 (X ∗ , x) → π1 (U ∗ , u) does not depend on the choice of x ∈ f −1 (u); we will call this image the subgroup of f and we will denote it by Γf .