arXiv subjects

M. Spivakovsky

Publications and source records attributed to M. Spivakovsky.

8 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 $$ \nu:K^*\twoheadrightarrow\Gamma $$ be a valuation centered at $R$ and let $R_\nu$ 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 $\nu$ by a suitable extension $\widehat\nu_-$ 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 $\widehat\nu_-$ 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 $\widehat\nu_-$ is a tight extension then its graded algebra is birational to that of $\nu$ (the converse is not known and might not be true). In particular, the value group of $\widehat\nu_-$ is $\Gamma$. 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↗

arXiv2022

On stable and fixed polynomials

Let $ν$ be a rank one valuation on $K[x]$ and $Ψ_n$ the set of key polynomials for $ν$ of degree $n\in\N$. We discuss the concepts of being $Ψ_n$-stable and $(Ψ_n,Q)$-fixed. We discuss when these two concepts coincide. We use this discussion to present a simple proof of Proposition 8.2 of [5] and Theorem 1.2 of [5].

math.AC↗

arXiv2020

On the structure of the graded algebra associated to a valuation

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra ${\rm gr}_v(R)$ of a subring $(R,\mathfrak{m})$ of a valuation ring $\mathcal{O}_v$, for which $Kv:=\mathcal{O}_v / \mathfrak{m}_v=R / \mathfrak{m}$, is isomorphic to $Kv[t^{v(R)}]$, where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist.

math.AC↗

arXiv2018

On the strong separation conjecture

This paper contains a partial result on the Pierce--Birkhoff conjecture on piece-wise polynomial functions defined by a finite collection {f 1,. .., f r} of polynomials. In the nineteen eighties, generalizing the problem from the polynomial ring to an artibtrary ring $Σ$, J. Madden proved that the Pierce--Birkhoff conjecture for $Σ$ is equivalent to a statement about an arbitrary pair of points $α$, $β$ $\in$ Sper $Σ$ and their separating ideal < $α$, $β$ >, we refer to this statement as the local Pierce-Birkhoff conjecture at $α$, $β$. In [8] we introduced a slightly stronger conjecture, also stated for a pair of points $α$, $β$ $\in$ Sper $Σ$ and the separating ideal < $α$, $β$ >, called the Connectedness conjecture, about a finite collection of elements {f 1, . . ., fr} $\subset$ $Σ$. In the paper [10] we introduced a new conjecture, called the Strong Connectednessconjecture, and proved that the Strong Connectedness conjecture in dimension n--1 implies the Strong Connectedness conjecture in dimension n in the case when ht(< $α$, $β$ >) $\le$ n -- 1.The Pierce-Birkhoff Conjecture for r = 2 is equivalent to the Connectedness Conjecture for r = 1, this conjecture is called the Separation Conjecture. The Strong Connectedness Conjecture for r = 1 is called the Strong Separation Conjecture. In the present paper, we fix a polynomial f $\in$ R[x, z] where R is a real closed field and x = (x1, . . ., xn), z are n + 1 independent variables. We define the notion of two points $α$, $β$ $\in$ Sper R[x, z] being in good position with respect to f. The main result of this paper is a proof of the Strong Separation Conjecture in the case when $α$ and $β$ are in good position with respect to f.

math.AG↗

arXiv2014

Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$ \mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; $$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. 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 $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ 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

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↗