arXiv subjects

Bernard Teissier

Publications and source records attributed to Bernard Teissier.

At least 19 recordsLinked to original sources

arXiv2025

On Kaplansky's Embedding Theorem

Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $ν$ be a zero dimensional valuation of rank one centered in $R$ with value group $Φ$. We show that there is a valuation preserving embedding of $R$ in the ring $k[[t^{Φ_{\geq 0}}]]$ of Hahn series. The method relies on torific local uniformization for rational valuations with finitely generated semigroup and the approximation of rational valuations by semivaluations with finitely generated semigroup.

math.AC↗

arXiv2023

Sur l'article On some ideals of differentiable functions

This text is a commentary on the paper "On some ideals of differentiable functions" of Ren{é} Thom which appeared in Volume II of his "Oeuvres Math{é}matiques" published by the Soci{é}t{é} math{é}matique de France, s{é}rie "Documents Math{é}matiques", in 2019.

math.HO↗

arXiv2023

Some ideas in need of clarification in resolution of singularities and the geometry of discriminants

This text, which appeared in volume 2313 of the Springer Lecture Notes in Math. dedicated to Catriona Byrne, presents problems in two different areas of algebraic geometry. The first concerns the role of ``infinitely singular'' or ``non Abhyankar'' valuations in the study of local uniformization of valuations with a view to resolution of singularities in positive characteristic. The second one concerns the relationship of the geometry of the discriminant of real miniversal unfoldings in the sense of Thom with the movements of Morse functions on a cobordism which differential geometers use.

math.AG↗

arXiv2023

About the work of Albert Lautman

Some reflections on the role in the development of Mathematics of our unconscious perception of the world and just as unconscious organizing pulsions for those perceptions.

math.HO↗

arXiv2021

On the construction of valuations and generating sequences on hypersurface singularities

Suppose that (K, $ν$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $ω$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ν$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $ω$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $ω$ as an extension of the associated graded ring of A for the filtration defined by $ν$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $ω$ A[z]/(f (z)) generate it as a gr $ν$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $ν$) $\subset$ (L, $ω$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $ν$ if and only if there is no defect. In this case, gr $ω$ A1[z]/(f (z)) is a finitely presented gr $ν$ A1-module. This is an improved version, thanks to a referee's remarks.

math.AG↗

arXiv2020

Lipschitz fractions of a complex analytic algebra and Zariski saturation

This text is the English translation, due to Naoufal Bouchareb, of an unpublished manuscript of 1969 (the French version is available on HAL as hal-00384928) inspired by Zariski's theory of saturation. Its publication is justified by the fact that it appears now as a precursor of the recently developed study of the Lipschitz geometry (for the outer metric) of germs of complex analytic spaces. This version contains some new footnotes and an additional bibliography.

math.AG↗

arXiv2019

Valuations and henselization

We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation $ν$ centered in a local domain R uniquely determines a minimal prime H($ν$) of the henselization R h of R and an extension of $ν$ centered in R h /H($ν$), which has the same value group as $ν$. Our method, which assumes neither that R is noetherian nor that it is integrally closed, is to reduce the problem to the extension of the valuation to a quotient of a standard {é}tale local R-algebra and in that situation to draw valuative consequences from the observation that the Newton-Hensel algorithm for constructing roots of polynomials produces sequences that are always pseudo-convergent in the sense of Ostrowski. We then apply this method to the study of the approximation of elements of the henselization of a valued field by elements of the field and give a characterization of the henselian property of a local domain (R, m R) in terms of the limits of certain pseudo-convergent sequences of elements of m R for a valuation centered in it. Another consequence of our work is to establish in full generality a bijective correspondence between the minimal primes of the henselization of a local domain R and the connected components of the Riemann-Zariski space of valuations centered in R.

math.AG↗

arXiv2018

Two points of the boundary of toric geometry

This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid some ramification problems arising in the study of local uniformization in positive characteristic, and the second one because it uses limits of projective systems of equivariant birational maps of toric varieties to study the space of additive preorders on ${\mathbf Z}^r$ for $r\geq 2$.

math.AG↗

arXiv2017

Local polar varieties in the geometric study of singularities

This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space $X$ into nonsingular strata with the property that the local geometry of $X$ is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of $X\subset \C^n$ of hyperplanes of $\C^n$ tangent to $X$ at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincaré characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Plücker-type formula for the degree of the dual of a projective variety. The degree of the dual of a projective variety is expressed in terms of Euler-Poincaré characteristics attached to the minimal Whitney stratification of the variety.

math.AG↗

arXiv2016

Overweight deformations of affine toric varieties and local uniformization

We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the valuation is finitely generated (which implies that the valuation is Abhyankar) the valuation can be uniformized in an embedded way by a birational map which is monomial with respect to a suitable system of generators of the maximal ideal. We prove that conversely if a valuation is Abhyankar after a birational modification and localization at the point picked by the valuation one obtains a ring whose semigroup of values is finitely generated. Combining the two results and using the good behavior of Abhyankar valuations with respect to composition and completion gives local uniformization for all Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. Some general results on Abhyankar valuations are by-products of the method of proof.

math.AG↗

arXiv2013

Toric Geometry and the Semple-Nash modification

This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part it is shown that over an algebraically closed base field of zero characteristic the Semple-Nash modification of a general toric variety is isomorphic to the blowing up of the sheaf of logarithmic jacobian ideals and that in any characteristic this blowing-up is an isomorphism if and only if the toric variety is non singular. In the second part we prove that orders on the lattice of monomials (toric valuations) of maximal rank are uniformized by iterated Sempla-Nash modifications.

math.AG↗

arXiv2012

Jacobian Newton Polyhedra and equisingularity

This is the LaTeX version of the handwritten notes of a lecture at the Kyoto Singularities Symposium held at the RIMS in April 1978. It presents the relationship of various invariants of isolated singularities of complex analytic hypersurfaces with the jacobian Newton polygon built from the polar curves. An appendix presents features of the semiring of Newton polygons, because the product of Jacobian Newton polygons corresponds to the Thom-Sebastiani sum of hypersurfaces. Analogies between mixed multiplicities and mixed volumes are noticed. They have been explained since.

math.AG↗

arXiv2012

Some resonances of Lojasiewicz inequalities

This note presents three resonances in commutative algebra and analytic geometry of the concept of Lojasiewicz inequality. The first is the interpretation in complex analytic geometry of the best possible exponent for a function g with respect to an ideal I at a point of a reduced complex space X as the inclination of a edge of a Newton polygon associated to the dicritical components of as log resolution of I. The second calls attention to recent results which show that some rational numbers connot be Lojasiewicz exponents for the gradient inequality of a holomorphic function of two variables. The last one reports on a recent result of Moret-Bailly which opens perspectives for a Lojasiewicz inequality in infinite dimensional spaces.

math.CV↗

arXiv2010

Formes de Whitney et primitives relatives de formes différentielles sous-analytiques

Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $ω$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $ω$ has a relative primitive: there is a subanalytic $(r-1)$-form $Ω$ such that $dgΛ(ω-dΩ)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of "relative Whitney forms" associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of François Treves.

math.AP↗

arXiv2008

Géométrie et cognition; l'exemple du continu

In this paper I propose the idea to establish a clear distinction between the foundations of truth and the foundations of meaning in Mathematics. I explore on the most basic example, the mathematical line, the possibility that the foundations of its meaning are provided by a protomathematical object resulting from the identification by our perceptual system of the visual line and the vestibular line, a point of view suggested by recent results of neurophysiology.

math.HO↗

arXiv2008

Semigroups of valuations on local rings, II

Given a noetherian local domain $R$ and a valuation $ν$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $Γ$ of $ν$. We show that this growth condition imposes restrictions on the semigroups $ν(R\setminus \{0\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \cite{C2} of the first author. Given an ordered embedding $Γ\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $ν$, we also study the shape in ${\mathbf R}^h$ of the parts of $Γ$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $ν(R\setminus \{0\})$.

math.CV↗

arXiv2008

Clôture intégrale des idéaux et équisingularité

This text has two parts; the first is the essentially unmodified text of the 1973-74 seminar of M. Lejeune-Jalabert and B. Teissier on integral dependence in complex analytic geometry with J-J. Risler's appendix on the Lojasiewicz exponents in the real analytic framework. The second part consists of seven complements written in 2007 surveying recent results directly connected to the content of the seminar. The main results of the first part concern the asymptotic order function with respect to an ideal and in particular its connection with the Lojasiewicz exponent. Another aspect concerns the finiteness properties of the graded algebra associated with the filtration by the asymptotic order function.

math.CV↗

arXiv2007

Semigroups of valuations on local rings

In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no constraints on the semigroup beyond those known classically.

math.AG↗