arXiv subjects

Mark Spivakovsky

Publications and source records attributed to Mark Spivakovsky.

At least 19 recordsLinked to original sources

arXiv2025

On defect in finite extensions of valued fields

In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify defectless fields and study properties related to defect. In this paper we study the relations between the following properties of valued fields: simply defectless, immediate-defectless and algebraically maximal. The main result of the paper is an example of an algebraically maximal field that admits a simple defect extension. For this, we introduce the notion of quasi-finite elements in the generalized power series field $k\left(\left(t^\Gamma\right)\right)$.

math.AC↗

arXiv2025

Valuation rings in simple algebraic extensions of valued fields

Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal O_L$ in terms of generators and relations over $\mathcal O_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of $\mathcal O_L$ over $\mathcal O_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of $\mathcal O_L$ over $\mathcal O_K$.

math.AC↗

arXiv2024

A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero Conjecture

The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $n$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $n$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $np^\ell$, $\ell\in \mathbb{N}$, where $p$ is a good prime for $n$. In this paper we give an explicit description of the set of bad primes in any given degree $n$. In particular, we show that if the conjecture holds in degree $n$ then the bad primes for $n$ are bounded above by $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.

math.AC↗

arXiv2024

Cusps of caustics by reflection in ellipses

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $\Gamma_n$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$. It is known that $\Gamma_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $\Gamma_n$, though we do not prove that these are the only cusps.

math.DG↗

arXiv2023

A note on the Casas-Alvero Conjecture

The Casas--Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\_i(f)$ is a power of a linear polynomial. Let $f=x^d+a\_1x^{d-1}+\cdots+a\_1x \in K[a\_1,\ldots,a\_{d-1}][x]$ and let $R\_i = Res(f,H\_i(f))\in K[a\_1,\ldots,a\_{d-1}]$ be the resultant of $f$ and $H\_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\_1,\ldots,R\_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R\_1,\ldots,R\_{d-1})=d-1$ in $K[a\_1,\ldots,a\_{d-1}]$. In this paper we prove a very partial result in this direction : if $i \in \{d-3,d-2,d-1\}$ then $R\_i \notin \sqrt{(R\_1,\ldots,\breve{R\_i},\ldots,R\_{d-1}}$.

math.AC↗

arXiv2023

K\"ahler differentials, pure extensions and minimal key polynomials

The main object of study in this paper is the module $\Omega$ of K\"ahler differentials of an extension of valuation rings. We show that in the case of pure extensions $\Omega$ has a very good description. Namely, it is isomorphic to the quotient of two modules defined by final segments in the value group of the valuation. This allows us to describe the annihilator of $\Omega$ and to give criteria for $\Omega$ to be finitely generated and to be finitely presented. We also discuss how to relate the module of K\"ahler differentials of pure extensions to minimal key polynomials.

math.AC↗

arXiv2023

On the set of bad primes in the study of Casas-Alvero Conjecture

The Casas-Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $d$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $d$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $dp^\ell$, $\ell\in\mathbb{N}$, where $p$ is a good prime for $d$. In this paper we calculate certain distinguished monomials appearing in the resultant $R(f,H_i(f))$ and obtain a (non-exhaustive) list of bad primes for every degree $d\in\mathbb{N}\setminus\{0\}$.

math.AC↗

arXiv2023

The module of K\"ahler differentials for extensions of valuation rings

The main goal of this paper is to characterize the module of K\"ahler differentials for an extension of valuation rings. More precisely, we consider a simple algebraic valued field extension $(L/K,v)$ and the corresponding valuation rings $\VR_L$ and $\VR_K$. In the case when $e(L/K,v)=1$ we present a characterization for $\Omega_{\VR_L/\VR_K}$ in terms of a given sequence of key polynomials for the extension. Moreover, we use our main result to present a characterization for when $\Omega_{\VR_L/\VR_K}=\{0\}$.

math.AC↗

arXiv2022

Graded rings associated to valuations and direct limits

In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.

math.AC↗

arXiv2020

On common extensions of valued fields

Given a valuation $v$ on a field $K$, an extension $\bar{v}$ to an algebraic closure and an extension $w$ to $K(X)$. We want to study the common extensions of $\bar{v}$ and $w$ to $\bar{K}(X)$. First we give a detailed link between the minimal pairs notion and the key polynomials notion. Then we prove that in the case when $w$ is a transcendental extension, then any sequence of key polynomials admits a maximal element, and in case this sequence does not contain a limit key polynomial, then any root of the last key polynomial, describe a common extension.

math.AC↗

arXiv2020

Newton non-degenerate $\mu$-constant deformations admit simultaneous embedded resolutions

Let $\mathbb{C}^{n+1}_o$ denote the germ of $\mathbb{C}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $\mathbb{C}^{n+1}_o$ and $W$ a deformation of $V$ over $\mathbb{C}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we will prove that $W$ is a $\mu$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(W(\mathbb{C}_o)_m)_{red} \rightarrow \mathbb{C}_{o}$ is flat, where $W(\mathbb{C}_o)_m$ is the relative space of $m$-jets. On the way tothe proof of our main result, we give a complete answer to a question ofArnold on the monotonicity of Newton numbers in the case of convenientNewton polyhedra.

math.AG↗

arXiv2019

Key Polynomials in dimension 2

Let $R$ be a two-dimensional regular local ring. In this paper, we prove that there is a bijection between the set of all valuations of $Quot(R)$ centered at $R$ and valuations of $k(x,y)$ centered at $k[x,y]_{(x,y)}$, where $k$ is the residue field of $R$ and $x$ and $y$ are independent variables. Moreover, we give a new proof, for the fact that the set of all normalized real valuations centered at $R$ admits a structure of non metric tree.

math.AG↗

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

Key polynomials and pseudo-convergent sequences

In this paper we introduce a new concept of key polynomials for a given valuation $ν$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of $ν$ associated to each key polynomial is a valuation. Moreover, we prove that every valuation $ν$ on $K[x]$ admits a sequence of key polynomials that completely determines $ν$ (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.

math.AC↗

arXiv2015

On the local uniformization problem

In this paper we give a short introduction to the local uniformization problem. This follows a similar line as the one presented by the second author in his talk at ALANT 3. We also discuss our paper on the reduction of local uniformization to the rank one case. In that paper, we prove that in order to obtain local uniformization for valuations centered at objects of a subcategory of the category of noetherian integral domains, it is enough to prove it for rank one valuations centered at objects of the same category. We also announce an extension of this work which was partially developed during ALANT 3. This extension says that the reduction mentioned above also works for noetherian rings with zero divisors (including the case of non-reduced rings).

math.AC↗

arXiv2015

Reduction of local uniformization to the case of rank one valuations for rings with zero divisors

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.

math.AC↗

arXiv2014

Equisingularity in one parameter families of generically reduced curves

We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.

math.AG↗

arXiv2014

The analogue of Izumi's Theorem for Abhyankar valuations

A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal.

math.AC↗