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Let $( A , m )$ be a Noetherian [[Local ring|local ring]] and $\hat{A}$ its completion. $A$ has the Artin approximation property (in brief, $A$ has AP) if every finite system of polynomial equations over $A$ has a solution in $A$ if it has one in $\hat{A}$. In fact, $A$ has the Artin approximation property if and only if for every finite system of polynomial equations $f$ over $A$ the set of its solutions in $A$ is dense, with respect to the $m$-adic topology, in the set of its solutions in $\hat{A}$. That is, for every solution $\widehat{y}$ of $f$ in $\hat{A}$ and every positive integer $c \in \bf N$ there exists a solution $y _ { c }$ of $f$ in $A$ such that $y _ { c } \cong \widehat { y }$ modulo $m ^ { c }\hat{ A}$. The study of Artin approximation started with the famous papers of M. Artin [[#References|[a3]]], [[#References|[a4]]], which state that the convergent power series rings over a non-trivial valued field of characteristic zero, the Henselization of a local ring essentially of finite type over a field, and an excellent Dedekind ring all have the Artin approximation property. The first result was extended by M. André [[#References|[a1]]] to certain convergent formal power series rings over a field of non-zero characteristic.

The following assertion holds: A Noetherian local ring has AP if and only if it is excellent and Henselian.

The necessity is stated in [[#References|[a24]]], a weaker result, namely that AP implies Henselian and universally Japanese, being proved in [[#References|[a14]]], (5.4), and [[#References|[a9]]]. The sufficiency gives a positive answer to Artin's conjecture [[#References|[a5]]] and is a consequence (see [[#References|[a21]]], (1.3), and [[#References|[a27]]]) of the following theorem on general Néron desingularization ([[#References|[a20]]], [[#References|[a21]]], [[#References|[a23]]], [[#References|[a2]]], [[#References|[a16]]], [[#References|[a27]]], [[#References|[a26]]]): A morphism $u : A \rightarrow A ^ { \prime }$ between Noetherian rings is regular (i.e. it is flat and for every field $K$ that is a finite $A$-algebra, the ring $K \otimes _ { A } A ^ { \prime }$ is regular) if and only if it is a filtered inductive limit of smooth algebras of finite type.

Roughly speaking, general Néron desingularization says in particular that if $u$ is a regular morphism of Noetherian rings, then every finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$ can be enlarged to a finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$, for which one may apply the implicit function theorem. Another consequence of general Néron desingularization says that a regular local ring containing a field is a filtered inductive limit of regular local rings essentially of finite type over $\mathbf{Z}$. This is a partial positive answer to the Swan conjecture and, using [[#References|[a15]]], proves the Bass–Quillen conjecture in the equicharacteristic case (see also [[#References|[a22]]], [[#References|[a27]]]).

Let $( A , m )$ be a Noetherian local ring. $A$ has the strong Artin approximation property (in brief, $A$ has SAP) if for every finite system of equations $f$ in $Y = ( Y _ { 1 } , \dots , Y _ { s } )$ over $A$ there exists a mapping $\nu :\mathbf{N} \rightarrow \mathbf{N}$ with the following property: If $\tilde { y } \in A ^ { s }$ satisfies $f (\tilde{y}) \cong 0$ modulo $m ^ { \nu ( c ) }$, $c \in \bf N$, then there exists a solution $y \in A ^ { S }$ of $f$ with $y \cong \widetilde{y}$ modulo $m ^ { c }$.

M. Greenberg [[#References|[a13]]] proved that excellent Henselian discrete valuation rings have the strong Artin approximation property and M. Artin [[#References|[a4]]] showed that the Henselization of a local ring which is essentially of finite type over a field has the strong Artin approximation property.

The following assertion is true: A Noetherian complete local ring $A$ has the strong Artin approximation property. In particular, $A$ has AP if and only if it has SAP. A special case of this is stated in [[#References|[a11]]], together with many other applications.

When $A$ contains a field, some weaker results were stated in [[#References|[a29]]], [[#References|[a30]]]. In the above form, the result appeared in [[#References|[a17]]], but the proof there has a gap in the non-separable case, which was repaired in [[#References|[a14]]], Chap. 2. In [[#References|[a8]]] it was noted that SAP is more easily handled using ultraproducts. Let $D$ be a non-principal [[Ultrafilter|ultrafilter]] on $\mathbf{N}$ (i.e. an ultrafilter containing the filter of [[cofinite subset]]s of $\mathbf{N}$). The ultraproduct $A ^ { * }$ of $A$ with respect to $D$ is the factor of $A ^ {\bf N }$ by the ideal of all $( a _ { n } ) _ { n \in \mathbf{N} }$ such that the set $\{ n : a _ { n } = 0 \} \in D$. Assigning to $a \in A$ the constant sequence $( a , a , \dots )$ one obtains a ring morphism $A \rightarrow A ^ { * }$. Using these concepts, easier proofs of the assertion were given in [[#References|[a19]]] and [[#References|[a10]]]. The easiest one is given in [[#References|[a21]]], (4.5), where it is noted that the separation $A _ { 1 } = A ^ { * } / \cap _ { i \in \mathbf{N} } m ^ { i } A ^ { * }$ of $A ^ { * }$ in the $m$-adic topology is Noetherian, that the canonical mapping $u : A \rightarrow A _ { 1 }$ is regular if $A$ is excellent and that $A$ is SAP if and only if for every finite system of polynomial equations $f$ over $A$, for every positive integer $c$ and every solution $\tilde{y}$ of $f$ in $A _ { 1 }$, there exists a solution $y _ { c }$ of $f$ in $A ^ { * }$ which lifts $\tilde{y}$ modulo $m ^ { c } A ^ { * }$. The result follows on applying general Néron desingularization to $u$ and using the implicit function theorem.

Theorems on Artin approximation have many direct applications in [[Algebraic geometry|algebraic geometry]] (for example, to the algebraization of versal deformations and the construction of algebraic spaces; see [[#References|[a6]]], [[#References|[a5]]]), in algebraic number theory and in commutative algebra (see [[#References|[a4]]], [[#References|[a14]]], Chaps. 5, 6). For example, if $( R , m )$ is a Noetherian complete local domain and $( a _ { i } ) _ { i \in \mathbf{N} }$ is a sequence of elements from $R$ converging to an irreducible element $a$ of $R$, then G. Pfister proved that $a_i$ is irreducible for $i \gg 1$ (see [[#References|[a14]]], Chap. 5). Using these ideas, a study of approximation of prime ideals in the $m$-adic topology was given in [[#References|[a18]]]. Another application is that the completion of an excellent Henselian local domain $A$ is factorial if and only if $\hat{A}$ is factorial [[#References|[a21]]], (3.4).

All these approximation properties were studied also for couples $( R , a )$, were $R$ is not necessarily local and $a$ is not necessarily maximal. A similar proof shows that the Artin approximation property holds for a Henselian couple $( R , a )$ if $R$ is excellent [[#References|[a21]]], (1.3). If $R / a$ is not Artinian, then $R _ { 1 } = R ^ { * } / \cap _ { i \in \mathbf{N} } a ^ { i } R ^ { * }$ is not Noetherian and SAP cannot hold in this setting, because one cannot apply general Néron desingularization. Moreover, the SAP property does not hold for general couples, as noticed in [[#References|[a25]]].

A special type of Artin approximation theory was required in singularity theory. Such types were studied in [[#References|[a14]]], Chaps. 3, 4. However, the result holds even in the following extended form: Let $( A , m )$ be an excellent Henselian local ring, $\hat{A}$ its completion, $A \langle  X \rangle $ the Henselization of $A[X]$, $X = ( X _ { 1 } , \ldots , X _ { n } )$, in $( X )$, $f$ a finite system of polynomial equations over $A \langle  X \rangle $ and $\hat { y } = ( \hat { y } _ { 1 } , \dots , \hat { y } _ { n } ) \in \hat { A } [ [ X ] ] ^ { n }$ a formal solution of $f$ such that $\hat { y } _ { i } \in \hat { A } [ [ X _ { 1 } , \dots , X _ { s _ { i } } ] ]$, $1 \leq i \leq n$, for some positive integers $s _ { i } \leq n$. Then there exists a solution $y = ( y _ { 1 } , \dots , y _ { n } )$ of $f$ in $A \langle  X \rangle $ such that $y _ { i } \in A \langle X _ { 1 } , \dots , X _ { s_i } \rangle$, $1 \leq i \leq n$, and $y _ { i } \cong \hat { y } _ { i }$ modulo $( m , X _ { 1 } , \dots , X _ { s_i } ) ^ { c }$, $1 \leq i \leq n$, for $c &gt; 1$.

The proof is given in [[#References|[a21]]], (3.6), (3.7), using ideas of H. Kurke and Pfister, who noticed that this assertion holds if $A ( X _ { 1 } , \dots , X _ { n } )$ has AP, where $A$ is an excellent Henselian local ring. If the sets of variables $X$ of $\hat{y}_ { i }$ are not "nested" (i.e. they are not totally ordered by inclusion), then the assertion does not hold, see [[#References|[a7]]]. If $A = \mathbf{C} \{ Z _ { 1 } , \dots , Z _ { r } \}$ is the convergent power series ring over $\mathbf{C}$ and the algebraic power series rings $A ( X _ { 1 } , \dots , X _ { s _ { i } } )$ are replaced by $A \{ X _ { 1 } , \dots , X _ { s _ { i } } \}$, then the theorem does not hold, see [[#References|[a12]]]. Extensions of this theorem are given in [[#References|[a28]]], [[#References|[a27]]].

====References====
<table>
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  <tr><td valign="top">[a2]</td> <td valign="top"> M. André, "Cinq exposés sur la desingularization" ''École Polytechn. Féd. Lausanne'' (1991) (Handwritten manuscript)</td></tr>
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</table>