arXiv subjects

Nathália Moraes de Oliveira

Publications and source records attributed to Nathália Moraes de Oliveira.

4 recordsLinked to original sources

arXiv2020

Computation of residual polynomial operators of inductive valuations

Let $(K,v)$ be a valued field, and $μ$ an inductive valuation on $K[x]$ extending $v$. Let $G_μ$ be the graded algebra of $μ$ over $K[x]$, and $κ$ the maximal subfield of the subring of $G_μ$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $κ$ and the residual polynomial operator $R_μ: K[x]\toκ[y]$, where $y$ is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.

math.AG↗

arXiv2019

Invariants of algebraic elements over henselian fields

Let (K,v) be a henselian valued field. In this paper, we use Okutsu sequences for monic, irreducible polynomials in K[x], and their relationship with MacLane chains of inductive valuations on K[x], to obtain some results on the computation of invariants of algebraic elements over K.

math.NT↗

arXiv2019

Defectless polynomials over henselian fields and inductive valuations

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on $\,\mathbb{P}^{dless}$, we establish a canonical bijection $\mathbb{M}\to \mathbb{P}^{dless}/\!\!\approx$, where $\mathbb{M}$ is a discrete MacLane space, constructed in terms of inductive valuations on $K[x]$ extending $v$.

math.AG↗

arXiv2016

Reduced normal form of local integral bases

We introduce a canonical form for reduced bases of integral closures of discrete valuation rings, and we describe an algorithm for computing a basis in reduced normal form. This normal form has the same applications as the Hermite normal form: identification of isomorphic objects, construction of global bases by patching local ones, etc. but in addition the bases are reduced, which is a crutial property for several important applications. Except for very particular cases, a basis in Hermite normal form cannot be reduced.

math.NT↗