Remark on a theorem of H. Hauser on textile maps
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.
math.AG↗
arXiv subjects
Guillaume Rond
Publications and source records attributed to Guillaume Rond.
Higher order approximation of analytic sets by topologically equivalent algebraic sets
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 analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.
math.CV↗
About the algebraic closure of the field of power series in several variables in characteristic zero
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 depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theorem for monic polynomials whose discriminant is weighted homogeneous.
math.AC↗
An irreducibility criterion for power series
We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous known results concerning quasi-ordinary polynomials.
math.CV↗
Local zero estimates and effective division in rings of algebraic power series
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 that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally we apply these results to prove a gap theorem for power series which are remainders of the Grauert-Hironaka-Galligo Division Theorem.
math.AC↗
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↗
Local topological algebraicity of analytic function germs
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) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.
math.AG↗
Bornes effectives des fonctions d'approximation des solutions formelles d'équations binomiales
The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded.
math.AC↗
Lojasiewicz inequality over the ring of power series in two variables
We prove a Lojasiewicz type inequality for a system of polynomial equations with coefficients in the ring of formal power series in two variables. This result is an effective version of the Strong Artin Approximation Theorem. From this result we deduce a bound of Artin functions of isolated singularities.
math.AC↗
Asymptotic behaviour of standard bases
We prove here that the elements of any standard basis of $I^n$, where $I$ is an ideal of a Noetherian local ring and $n$ is a positive integer, have order bounded by a linear function in $n$. We deduce from this that the elements of any standard basis of $I^n$ in the sense of Grauert-Hironaka, where $I$ is an ideal of the ring of power series, have order bounded by a polynomial function in $n$.
math.AC↗
Homomorphisms of local algebras in positive characteristic
We investigate some properties of regularity of homomorphisms of local algebras over positive characteristic fields. We state a result of monomialization of such a homomorphism between algebras of analytic or algebraic power series. From this we deduce some extensions in positive characteristic of results due to S. Izumi and A. Gabrielov.
math.AC↗
The Abhyankar-Jung Theorem
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] $, $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero $\K$, and satisfying $a_1=0$, is $ν$-quasi-ordinary. That means that if the discriminant $Δ_P \in \K[[X]]$ is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of $\K[[X]]$ and the function germs of quasi-analytic families.
math.AC↗
An algebraic proof of Gabrielov's theorem about analytic homomorphisms in any characteristic
The proof of proposition 3.6 is not correct
math.AC↗
Sur la linearite de la fonction de Artin
We give here a counter-example to a conjecture of Spivakovsky. M. Spivakovsky conjectured that the function that appears in the strong Artin approximation theorem is bounded by a linear function. First we show that there is no Liouville theorem for the field of fractions of the ring of power series in several variables. We deduce from this example that there is no theory of elimination of quantifiers for the field of fractions of the ring of power series in several variables.
math.AC↗
Lemme d'Artin-Rees, theoreme d'Izumi et fonction de Artin
We interpret the Artin-Rees lemma and the Izumi theorem in term of Artin function and we obtain a stable version of the Artin-Rees lemma. We present different applications of these interpretations. First we show that the Artin function of $X_1X_2-X_3X_4$, as a polynomial in the ring of power series in more than three variables, is not bounded by an affine function. Then we prove that the Artin functions of a class of polynomials are bounded by affine functions and we use this to compute approximated integral closures of ideals.
math.AC↗
Series de Poincare motiviques geometrique et arithmetique d'un germe d'hypersurface irreductible quasi-ordinaire
We give here a description of the motivic Poincare series in case of irreducible quasi-ordinary hypersurfaces in all dimension. We give an explicit formula in a particular case. Finally, for such singularities, we give a constructive proof of a result of rationnality of such series proved by Denef and Loeser in all generality.
math.AG↗
Approximation diophantienne dans les corps de series en plusieurs variables
We give here a result of diophantine approximation between $Ø_N$, the ring of power series in several variables, and the completion of the valuation ring that dominates $Ø_N$ for the $\m$-adic topology. We deduce from this that the Artin function of a homogenous polynomial in two variables is bounded by an affine function, which can be interpreted in term of a Lojasiewicz inequality on the wedges space.
math.AG↗
Theoreme d'approximation cylindrique dans le cas des anneaux analytiques
This paper has been withdrawn by the authors because M. Aschenbrenner pointed out that the proof of theorem 3.1 was incorrect.
math.AC↗