arXiv subjects

Jean-Philippe Monnier

Publications and source records attributed to Jean-Philippe Monnier.

14 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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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.

math.AG↗

arXiv2021

Central Algebraic Geometry and Seminormality

We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties. We provide a construction of the central seminormalization by a decomposition theorem in elementary central gluings. The existence of a central seminormalization is established in the affine case and for real schemes.

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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

Semi-algebraic geometry with rational continuous functions

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.

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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

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

Very special divisors on 4-gonal real algebraic curves

Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.

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arXiv2009

Very special divisors on real algebraic curves

We study special linear systems called "very special" whose dimension does not satisfy a Clifford type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of very special linear systems that are simple are also given.

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arXiv2006

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve.

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arXiv2005

Clifford Theorem for real algebraic curves

We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

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