arXiv subjects

M. A. Olalla Acosta

Publications and source records attributed to M. A. Olalla Acosta.

9 recordsLinked to original sources

arXiv2026

Extending valuations of local domains to complete local domains without changing the value group

Let $(R,m,k)$ be an excellent local noetherian domain with field of fractions $K$. Let $$ ν:K^*\twoheadrightarrowΓ$$ be a valuation centered at $R$ and let $R_ν$ be the corresponding valuation ring of $K$, dominating $R$. Denote by $\widehat R$ the $m$-adic completion of $R$. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace $R$ by its $m$-adic completion $\widehat R$ and $ν$ by a suitable extension $\widehatν_-$ to $\frac{\widehat R}P$ for a suitably chosen prime ideal $P$, such that $$ P\cap R=(0). $$ In a previous article we gave a systematic description of all such extensions $\widehatν_-$ and defined the notion of tight extensions that are of particular interest for applications (see Herrera, Olalla, Spivakovsky and Teissier, Extending a valuation centered in a local domain to its formal completion, Proc. London Math. Soc. (3) 105 (2012) 571--621). If $\widehatν_-$ is a tight extension then its graded algebra is birational to that of $ν$ (the converse is not known and might not be true). In particular, the value group of $\widehatν_-$ is $Γ$. The existence of tight extensions was conjectured by the last author (see Teissier, Valuations, deformations, and toric geometry, Fields Institute Communications, 33, 2003, 361-459). In the present paper we give a proof of Teissier's conjecture. An intended application of this result is an important step in two recent approaches to local uniformization in positive characteristic.

math.AC↗

arXiv2026

Key polynomials for simple extensions of valued fields

Let $ι:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $ν$ of rank 1. That is, we assume given a rank 1 valuation $ν$ of $K$ and its extension $ν'$ to $L$. Let $(R_ν,M_ν,k_ν)$ denote the valuation ring of $ν$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaquié, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $ι$ a countable well ordered set $$ \mathbf{Q}=\{Q_i\}_{i\inΛ}\subset K[x]; $$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $β_i=ν'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_ν=0$ then the set of key polynomials has order type at most $ω$, while in the case $\operatorname{char}\ k_ν=p>0$ this order type is bounded above by $ω\timesω$, where $ω$ stands for the first infinite ordinal.

math.AG↗

arXiv2012

Extending valuations to formal completions

This paper is an extended version of the talk given by Miguel Olalla at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper "Extending a valuation centered in a local domain to the formal completion" (Proc. London Math. Soc. (2012) 105 (3), 571-621) without grinding through all of its technical details.

math.AG↗

arXiv2012

Extending a valuation centered in a local domain to the formal completion

Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the m-adic completion of R. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace R by its m-adic completion \hat{R} and v by a suitable extension \hat{v} to \hat{R}/P for a suitably chosen prime ideal P, such that P \cap R = (0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions \hat{v} and to identify certain classes of extensions which are of particular interest for applications.

math.AG↗

arXiv2006

Rank one discrete valuations of power series fields

In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the Zorn's lemma or to know explicitly a section $σ$ of the natural homomorphism $R_v\to\d$ between the ring and the residue field of the valuation $v$. However, as a consequence of this construction, in section 7, we prove that $k((\X))$ can be embedded into a field $L((\Y))$, where $L$ is an algebraic extension of $k$ and the {\em ``extended valuation'' is as close as possible to the usual order function}.

math.AC↗

arXiv2006

Valuations in algebraic field extensions

Let $K\to L$ be an algebraic field extension and $ν$ a valuation of $K$. The purpose of this paper is to describe the totality of extensions $\left\{ν'\right\}$ of $ν$ to $L$ using a refined version of MacLane's key polynomials. In the basic case when $L$ is a finite separable extension and $rk ν=1$, we give an explicit description of the limit key polynomials (which can be viewed as a generalization of the Artin--Schreier polynomials). We also give a realistic upper bound on the order type of the set of key polynomials. Namely, we show that if $char K=0$ then the set of key polynomials has order type at most $\mathbb N$, while in the case $char K=p>0$ this order type is bounded above by $([\log_pn]+1)ω$, where $n=[L:K]$. Our results provide a new point of view of the the well known formula $\sum\limits_{j=1}^se_jf_jd_j=n$ and the notion of defect.

math.AC↗

arXiv2002

Valuations in fields of power series

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.

math.AC↗

arXiv2002

Rank one discrete valuations of $k((X_1,...X_n))$

In this paper we study the rank one discrete valuations of $k((X_1,... ,X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal $(\X)$. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the Zorn's lemma or to know explicitly a section $σ$ of the natural homomorphism $R_v\to\d$ between the ring and the residue field of the valuation $v$. However, as a consequence of this construction, in section 7, we prove that $k((\X))$ can be embedded into a field $L((\Y))$, where the {\em ``extended valuation'' is as close as possible to the usual order function}.

math.AC↗

arXiv2000

Computing the rational torsion of an elliptic curve using Tate normal form

It is a classical result (apparently due to Tate) that all elliptic curves with a torsion point of order n ($4 \leq n \leq 10$, or n = 12) lie in a one-parameter family. However, this fact does not appear to have been used ever for computing the torsion of an elliptic curve. We present here a extremely down-to-earth algorithm using the existence of such a family.

math.AG↗