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Goulwen Fichou

Publications and source records attributed to Goulwen Fichou.

At least 19 recordsLinked to original sources
arXiv2026

Real radiciality and monoreal extensions

We study irreducible polynomials admitting a single real root in any real closed field extension of the base field, called monoreal polynomials. We show some stability properties satisfied by the induced monoreal field extensions, and define the monoreal closure of a field. We make the link with a notion of real radiciality for ring extensions, and an injectivity property at the real spectrum level. We end with a geometric application, showing that injectivity implies surjectivity for the real spectrum mapping, under certain assumptions.

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arXiv2024

Motivic, logarithmic, and topological Milnor fibrations

We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato-Nakayama log-space, and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. As an application, we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of $\mathbb{R}_{>0}$, and show that it determines the classical motivic Milnor fibre. We also give precise formulae expressing how the introduced objects change under blowings-up. As an application, we show that the motivic Milnor fibre is well-defined as an element of a suitable Grothendieck ring without requiring that the Lefschetz motive be invertible.

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arXiv2024

The Geometry of Locally Bounded Rational Functions

This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of Łojasiewicz's inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.

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arXiv2023

On central orderings

We define the notion of central orderings for a general commutative ring $A$ which generalizes the notion of central points of irreducible real algebraic varieties. We study a central and a precentral loci which both live in the real spectrum of the ring $A$ and allow to state central Positivestellensätze in the spirit of Hilbert 17th problem.

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arXiv2022

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case, whereas it comes from the theory of real closed fields otherwise.

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arXiv2020

Motivic integration and Milnor fiber

We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called "nonarchimedean Milnor fiber" that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. We also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert, Loeser, and Merle. Finally, applying $T$-convex integration after descent, matching the Euler Characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries.

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arXiv2019

Weak and semi normalization in real algebraic geometry

We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and provide a full description of centrally seminormal curves.

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arXiv2018

Integral Closures In Real Algebraic Geometry

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.

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arXiv2018

Substitution Property for the Ring of Continuous Rational Functions

We study the substitution property for the ring R 0 (V) of continuous rational functions on a real algebraic affine variety V. We show that R 0 (V) satisfies a substitution property along points; moreover, when V is non-singular, it satisfies also a substitution property along Puiseux arcs, which characterizes R 0 (V).

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arXiv2017

On regulous and regular images of Euclidean spaces

In this work we compare the semialgebraic subsets that are images of regulous maps with those that are images of regular maps. Recall that a map f : R n $\rightarrow$ R m is regulous if it is a rational map that admits a continuous extension to R n. In case the set of (real) poles of f is empty we say that it is regular map. We prove that if S $\subset$ R m is the image of a regulous map f : R n $\rightarrow$ R m , there exists a dense semialgebraic subset T $\subset$ S and a regular map g : R n $\rightarrow$ R m such that g(R n) = T. In case dim(S) = n, we may assume that the difference S \ T has codimension $\ge$ 2 in S. If we restrict our scope to regulous maps from the plane the result is neat: if f : R 2 $\rightarrow$ R m is a regulous map, there exists a regular map g : R 2 $\rightarrow$ R m such that Im(f) = Im(g). In addition, we provide in the Appendix a regulous and a regular map f, g : R 2 $\rightarrow$ R 2 whose common image is the open quadrant Q := {x > 0, y > 0}. These maps are much simpler than the best known polynomial maps R 2 $\rightarrow$ R 2 that have the open quadrant as their image.

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arXiv2017

On relative grothendieck rings and algebraically constructible functions

We investigate Grothendieck rings appearing in real geometry, notably for arc-symmetric sets, and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by McCrory and Parusinski. We study in particular the duality and link operators, including its behaviour with respect to motivic Milnor fibres with signs.

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arXiv2015

Artin approximation compatible with a change of variables

We propose a version of the classical Artin approximation which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This results is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global version of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps.

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arXiv2014

Continuous functions in the plane regular after one blowing-up

We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these functions correspond to rational functions which become regular after some blowings-up, we work on the plane where it suffices to blow-up points and then we can count the number of stages of blowings-up necessary. In the latest parts of the paper, we investigate the ring of rational continuous functions on the plane regular after one stage of blowings-up. In particular, we prove a Positivstellensatz without denominator in this ring.

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arXiv2014

Real Milnor Fibres and Puiseux Series

Given a real polynomial function and a point in its zero locus, we defined a set consisting of algebraic real Puiseux series naturally attached to these data. We prove that this set determines the topology and the geometry of the real Milnor fibre of the function at this point. To achieve this goal, we balance between the tameness properties of this set of Puiseux series, considered as a real algebraic object over the field of algebraic Puiseux series, and its behaviour as an infinite dimensional object over the real numbers.

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arXiv2014

Fonctions Régulues

We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstelensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.

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arXiv2013

Continuous mappings between spaces of arcs

A blow-analytic homeomorphism is an arc-analytic subanalytic homeomorphism, and therefore it induces a bijective mapping between spaces of analytic arcs. We tackle the question of the continuity of this induced mapping between the spaces of arcs, giving a positive and a negative answer depending of the topology involved. We generalise the result to spaces of definable arcs in the context of o-minimal structures, obtaining notably a uniform continuity property.

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arXiv2012

Virtual Poincaré polynomial of the link of a real algebraic variety

The Euler characteristic of the link of a real algebraic variety is an interesting topological invariant in order to discuss local topological properties. We prove in the paper that an invariant stronger than the Euler Characteristic is well defined for the link of an algebraic variety: its virtual Poincaré polynomial.

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