7 citations · 15 across the 15 of their papers we have counts for
26 papers
Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent
Guillaume Rond
We answer in the negative the long-standing open question of whether biholomorphic equivalence implies algebraic equivalence for germs of real algebraic manifolds in .…
On the Nash points of subanalytic sets
Andre Belotto da Silva, Octave Curmi, Guillaume Rond
Based on a recently developed rank Theorem for Eisenstein power series, we provide new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or…
On rank Theorems for morphisms of local rings
André Belotto da Silva, Octave Curmi, Guillaume Rond
We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the families of complex analytic functions a…
A proof of A. Gabrielov's rank Theorem
André Belotto da Silva, Octave Curmi, Guillaume Rond
This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we dev…
Artin approximation over Banach spaces
Guillaume Rond
We give examples showing that the usual Artin Approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach alg…
Finiteness results concerning algebraic power series
Fuensanta Aroca, Julie Decaup, Guillaume Rond
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we…
Preordered groups and valued fields
Julie Decaup, Guillaume Rond
We study algebraic, combinatorial and topological properties of the set of preorders on a group, and the set of valuations on a field. We show strong analogies between these two ki…
Transcendental holomorphic maps between real algebraic manifolds in a complex space
Guillaume Rond
We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is c…
The minimal cone of an algebraic Laurent series
Fuensanta Aroca, Julie Decaup, Guillaume Rond
We study the algebraic closure of , the field of power series in several indeterminates over a field . In characteristic zero we show that the elemen…
Algebraic varieties are homeomorphic to varieties defined over number fields
Adam Parusinski, Guillaume Rond
We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers…
Multiparameter perturbation theory of matrices and linear operators
Adam Parusinski, Guillaume Rond
We show that a normal matrix with coefficient in , , can be diagonalized, provided the discriminant of its characteristic polynomia…
Support of Laurent series algebraic over the field of formal power series
Fuensanta Aroca, Guillaume Rond
This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our fir…
Remarks on Artin approximation with constraints
Dorin Popescu, Guillaume Rond
We study various approximation results of solutions of equations where and and are two sets of variables, and where some component…
Remark on a theorem of H. Hauser on textile maps
Guillaume Rond
We give a counter example to the new theorem that appeared in the survey \cite{H} on Artin approximation. We then provide a correct statement and a proof of it.
Local topological algebraicity with algebraic coefficients of analytic sets or functions
Guillaume Rond
We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.
An irreducibility criterion for power series
Guillaume Rond, Bernd Schober
We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous known results concerning quasi-ordinary polynomials.
Higher order approximation of analytic sets by topologically equivalent algebraic sets
Marcin Bilski, Krzysztof Kurdyka, Adam Parusinski +1
It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the an…
Linear nested Artin approximation theorem for algebraic power series
Francisco-Jesus Castro-Jiménez, Dorin Popescu, Guillaume Rond
We give a new and elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subri…
Artin Approximation
Guillaume Rond
In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result…
Local zero estimates and effective division in rings of algebraic power series
Guillaume Rond
We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show tha…
Local topological algebraicity of analytic function germs
Marcin Bilski, Adam Parusinski, Guillaume Rond
T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analyti…
About the algebraic closure of the field of power series in several variables in characteristic zero
Guillaume Rond
We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields de…
Asymptotic behaviour of standard bases
Guillaume Rond
We prove here that the elements of any standard basis of , where is an ideal of a Noetherian local ring and is a positive integer, have order bounded by a linear funct…
The Abhyankar-Jung Theorem
Adam Parusinski, Guillaume Rond
We show that every quasi-ordinary Weierstrass polynomial $P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] $, , over an algebraically closed field of characte…