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La conjecture de Green générique

Arnaud Beauville (2003/2004)

Séminaire Bourbaki

Une courbe C projective et lisse de genre g , non hyperelliptique, admet un plongement canonique dans un espace projectif ℙ g - 1 . Un résultat classique affirme que l’idéal gradué I C des équations de C dans ℙ g - 1 est engendré par ses éléments de degré 2 , sauf si C admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de I C en fonction de l’existence de systèmes linéaires spéciaux sur C . Claire Voisin vient de...

Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

Let F ( X ) = ∑ n ≥ 0 ( - 1 ) ε ₙ X - λ ₙ be a real lacunary formal power series, where εₙ = 0,1 and λ n + 1 / λ ₙ > 2 . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that Q ω ( X ) is a polynomial if and only if ω ∈ ℤ. In all the other cases Q ω ( X ) is an infinite formal power series; we discuss its algebraic...

Langage de Łukasiewicz et diagonales de séries formelles

Isabelle Fagnot (1996)

Journal de théorie des nombres de Bordeaux

Dans un corps fini, toute série formelle algébrique en une indéterminée est la diagonale d'une fraction rationnelle en deux indéterminées (Furstenberg 67). Dans cet article, nous donnons une nouvelle preuve de ce résultat, par des méthodes purement combinatoires.

Large superdecomposable E(R)-algebras

Laszlo Fuchs, Rüdiger Göbel (2005)

Fundamenta Mathematicae

For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands ≠ 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module, A is multiplication by an element of, A.

Laskerian lattices

C. Jayaram (2003)

Czechoslovak Mathematical Journal

In this paper we investigate prime divisors, B w -primes and z s -primes in C -lattices. Using them some new characterizations are given for compactly packed lattices. Next, we study Noetherian lattices and Laskerian lattices and characterize Laskerian lattices in terms of compactly packed lattices.

Lattice of ℤ-module

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9]....

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