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Division dans l'anneau des séries formelles à croissance contrôlée. Applications

Augustin Mouze (2001)

Studia Mathematica

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove a Weierstrass-Hironaka division theorem for such subrings. Moreover, given an ideal ℐ of A and a series f in A we prove the existence in A of a unique remainder r modulo ℐ. As a consequence, we get a new proof of the noetherianity of A.

Elasticity of factorizations in atomic monoids and integral domains

Franz Halter-Koch (1995)

Journal de théorie des nombres de Bordeaux

For an atomic domain R , its elasticity ρ ( R ) is defined by : ρ ( R ) = sup { m / n u 1 ⋯ u m = v 1 ⋯ v n for irreducible u j , v i ∈ R } . We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants μ m ( R ) defined by : μ m ( R ) = sup { n u 1 ⋯ u m = u 1 ⋯ v n for irreducible u j , v i ∈ R } . As a main result we characterize all orders in algebraic number fields having finite elasticity. On the way, we obtain a series of results concerning the invariants μ m and ρ for monoids and integral domains which are of independent interest.

Factorization in Krull monoids with infinite class group

Florian Kainrath (1999)

Colloquium Mathematicae

Let H be a Krull monoid with infinite class group and such that each divisor class of H contains a prime divisor. We show that for each finite set L of integers ≥2 there exists some h ∈ H such that the following are equivalent: (i) h has a representation h = u 1 · . . . · u k for some irreducible elements u i , (ii) k ∈ L.

Finiteness results for Abelian tree models

Jan Draisma, Rob H. Eggermont (2015)

Journal of the European Mathematical Society

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant§ refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure....

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