Artin approximation over Banach spaces

Linear nested Artin approximation theorem for algebraic power series

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 subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the com-mutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals.

Artin Approximation

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 and a similar result of Płoski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem, aimed at non-experts.

关于 Artin 逼近的进一步结果：针对映射芽 Maps(X,Y) 上的群作用及映射的 quiver

Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps

本文研究了具有任意奇点的概形芽映射 Maps(X,Y) 的 Artin 逼近，建立了强 Artin 逼近与 Płoski 逼近，并扩展至有根有向树的映射芽 quiver 情形。

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.

Chow 群：结构定理、无分母 RIEMANN-ROCH 与 ARTIN 逼近

Chow Groups: A Structure Theorem, RIEMANN-ROCH without denominators and ARTIN approximation

本文研究分歧正则局部环上的 Chow 群问题，给出该类环的刻画，证明特定开子集上 $(i-1)!\mathbb{A}^i(U)=0$，并建立 Hensel 化与完备化 Chow 群之间的构造性关系。

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.

完美域上的相对逼近次数与亨塞尔理性问题

Relative approximation degrees and the henselian rationality problem over perfect fields

本文研究完美域上的亨塞尔理性问题，完善相对逼近次数理论，推广亨塞尔次数界，分析Artin–Schreier归约，证明特定条件下亨塞尔理性可下传至基域K。

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.

On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra $\mathcal{A}({\bf{p}})$ of entire functions are established. For a given fixed sequence $(\bf{p}(n))_{n\geq 0}$ of positive real numbers, such that $\lim_{n\rightarrow \infty} {\bf{p}}(n)^{\frac{1}{n}}=\infty$, the Banach algebra $\mathcal{A}({\bf{p}})$ is the set of all entire functions $f$ such that $f(z)=\sum_{n=0}^\infty \hat{f}(n) z^n $ ($z\in \mathbb{C}$), where the sequence $(\hat{f}(n))_{n\geq 0}$ of Taylor coefficients of $f$ satisfies $\hat{f}(n)=O({\bf{p}}(n)^{-1})$ for $n\rightarrow \infty$, with pointwise add…

代数态射与弱有限奇点类型解析态射的左右逼近结果

Results on left-right approximation for algebraic morphisms and for analytic morphisms of weakly finite singularity type

本文将 Artin 逼近的左右版本（LRAP）及其 Płoski 强版本扩展至任意特征的代数与解析概形芽态射，引入了“弱有限奇点类型”态射类，并证明了其相关的高阶临界轨迹基本性质。

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.