arXiv subjects

Julie Decaup

Publications and source records attributed to Julie Decaup.

7 recordsLinked to original sources

arXiv2021

The minimal cone of an algebraic Laurent series

We study the algebraic closure of $\mathbb K(\!(x)\!)$, the field of power series in several indeterminates over a field $\mathbb K$. In characteristic zero we show that the elements algebraic over $\mathbb K(\!(x)\!)$ can be expressed as Puiseux series such that the convex hull of its support is essentially a polyhedral rational cone, strengthening the known results. In positive characteristic we construct algebraic closed fields containing the field of power series and we give examples showing that the results proved in characteristic zero are longer valid in positive characteristic.

math.AC↗

arXiv2021

Some algebraic and topological properties of subspaces of (pre)orders

We study algebraic and topological properties of subsets of preorders on a group. In particular we study properties of the composition of two preorders, generalize a topological theorem of \cite{S} in the case of standard orders and show the same theorem in the case of standard preorders. We also show a property of standard valuations.

math.GR↗

arXiv2020

Simultaneous Monomialization

We give a new proof of the simultaneous embedded local uniformization Theorem in zero characteristic for essentially of finite type rings and for quasi excellent rings. The results are a consequence of the simultaneaous monomialization presented here. The methods develop the key elements theory that is a more subtle notion than the notion of key polynomials.

math.AC↗

arXiv2020

Finiteness results concerning algebraic power series

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power series of bounded complexity lying on an algebraic variety defined over the field of power series.

math.AC↗

arXiv2019

Preordered groups and valued fields

We study algebraic, combinatorial and topological properties of the set of preorders on a group, and the set of valuations on a field. We show strong analogies between these two kinds of sets and develop a dictionary for these ones. Among the results we make a detailed study of the set of preorders on $\mathbb Z^n$. We also prove that the set of valuations on a countable field of transcendence degree at least 2 is an ultrametric Cantor set.

math.GR↗

arXiv2016

Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane -- Vaquie

Let $ι:(K,ν)\hookrightarrow(K(x),μ)$ be a simple purely transcendental extension of valued fields. In order to study such an extension, M. Vaquié, generalizing an earlier construction of S. Mac Lane, introduced the notion of Key polynomials. In this paper we define a related notion of \textbf{abstract key polynomials} associated to $ι$ and study the relationship between them and key polynomials of Mac Lane -- Vaquié. Associated to each abstract key polynomial $Q$, we define the truncation $μ_{Q}$ of $μ$ with respect to $Q$ and we study the properties of those truncations. Roughly speaking, $μ_{Q}$ is an approximation to $μ$ defined by the key polynomial $Q$. We also define the notion of an abstract key polynomial $Q'$ being an \textbf{immediate successor} of another abstract key polynomial $Q$ (in this situation we write $Q<Q'$). The main comparison results proved in this paper are as follows:(1): An abstract key polynomial for $μ$ is a Mac Lane -- Vaquié key polynomial for the truncated valuation $μ_{Q}$.(2): If $Q<Q'$ are two abstract key polynomials for $μ$ then $Q'$ is a Mac Lane -- Vaquié key polynomial for $μ_{Q}$. (3) which, for a monic polynomial $Q\in K[x]$ and a valuation $μ'$ of $K(x)$, gives a sufficient condition for $Q$ to be an abstract key polynomial for $μ'$. Combined with an earlier result of M. Vaquié, this describes a class of pairs of valuations $(μ,μ')$ such that $Q$ is a Mac Lane -- Vaquié key polynomial for $μ$ and an abstract key polynomial for $μ'$.

math.AC↗

arXiv2016

Affine lines in the complement of a smooth plane conic

We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular auto-morphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of Q. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of Q over a field of arbitrary characteristic.

math.AG↗