# Difference between revisions of "Artin approximation"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026098.png" /> be an excellent Henselian local ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026099.png" /> its completion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260100.png" /> the Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260102.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260104.png" /> a finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260106.png" /> a formal solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260109.png" />, for some positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260110.png" />. Then there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260112.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260115.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260116.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260118.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260119.png" />. | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026098.png" /> be an excellent Henselian local ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026099.png" /> its completion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260100.png" /> the Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260102.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260104.png" /> a finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260106.png" /> a formal solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260109.png" />, for some positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260110.png" />. Then there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260112.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260115.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260116.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260118.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260119.png" />. | ||

− | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260120.png" /> has AP, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260121.png" /> is an excellent Henselian local ring. If the sets of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260123.png" /> are not | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260120.png" /> has AP, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260121.png" /> is an excellent Henselian local ring. If the sets of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260123.png" /> are not "nested" (i.e. they are not totally ordered by inclusion), then the assertion does not hold, see [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260124.png" /> is the convergent power series ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260125.png" /> and the algebraic power series rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260126.png" /> are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260127.png" />, then the theorem does not hold, see [[#References|[a12]]]. Extensions of this theorem are given in [[#References|[a28]]], [[#References|[a27]]]. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. André, "Artin's theorem on the solution of analytic equations in positive characteristic" ''Manuscripta Math.'' , '''15''' (1975) pp. 314–348 {{MR|379493}} {{ZBL|}} </TD></TR><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><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Artin, "On the solution of analytic equations" ''Invent. Math.'' , '''5''' (1968) pp. 277–291 {{MR|0232018}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Artin, "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58 {{MR|0268188}} {{ZBL|0181.48802}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Artin, "Construction techniques for algebraic spaces" , ''Actes Congres Internat. Math.'' , '''1''' (1970) pp. 419–423 {{MR|0427316}} {{ZBL|0232.14003}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Artin, "Versal Deformations and Algebraic Stacks" ''Invent. Math.'' , '''27''' (1974) pp. 165–189 {{MR|0399094}} {{ZBL|0317.14001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Becker, "A counterexample to Artin approximation with respect to subrings" ''Math. Ann.'' , '''230''' (1977) pp. 195–196 {{MR|0480508}} {{ZBL|0359.13007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Becker, J. Denef, L. Lipshitz, L. van den Dries, "Ultraproducts and approximation in local rings I" ''Invent. Math.'' , '''51''' (1979) pp. 189–203 {{MR|0528023}} {{ZBL|0416.13004}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Cipu, D. Popescu, "Some extensions of Néron's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260128.png" />-desingularization and approximation" ''Rev. Roum. Math. Pures Appl.'' , '''24''' : 10 (1981) pp. 1299–1304</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J. Denef, L. Lipshitz, "Ultraproducts and approximation in local rings II" ''Math. Ann.'' , '''253''' (1980) pp. 1–28 {{MR|0594530}} {{ZBL|0426.13010}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R. Elkik, "Solutions d'equations à coefficients dans une anneau (!!) henselien" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''6''' (1973) pp. 533–604 {{MR|345966}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A.M. Gabrielov, "The formal relations between analytic functions" ''Funkts. Anal. Prilozh.'' , '''5''' : 4 (1971) pp. 64–65 (In Russian) {{MR|0302930}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> M. Greenberg, "Rational points in Henselian discrete valuation rings" ''Publ. Math. IHES'' , '''31''' (1966) pp. 59–64 {{MR|0207700}} {{MR|0191897}} {{ZBL|0146.42201}} {{ZBL|0142.00901}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> H. Kurke, T. Mostowski, G. Pfister, D. Popescu, M. Roczen, "Die Approximationseigenschaft lokaler Ringe" , ''Lecture Notes Math.'' , '''634''' , Springer (1978) (Note: The proof of (3.1.1) is wrong) {{MR|0485851}} {{ZBL|0401.13013}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> H. Lindel, "On the Bass–Quillen conjecture concerning projective modules over polynomial rings" ''Invent. Math.'' , '''65''' (1981) pp. 319–323 {{MR|0641133}} {{ZBL|0477.13006}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> T. Ogoma, "General Néron desingularization based on the idea of Popescu" ''J. Algebra'' , '''167''' (1994) pp. 57–84 {{MR|1282816}} {{ZBL|0821.13003}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> G. Pfister, D. Popescu, "Die strenge Approximationseigenschaft lokaler Ringe" ''Invent. Math.'' , '''30''' (1975) pp. 145–174 {{MR|0379490}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> G. Pfister, D. Popescu, "Die Approximation von Primidealen" ''Bull. Acad. Polon. Sci.'' , '''27''' (1979) pp. 771–778 {{MR|603146}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Popescu, "Algebraically pure morphisms" ''Rev. Roum. Math. Pures Appl.'' , '''26''' : 6 (1979) pp. 947–977 {{MR|0546539}} {{ZBL|0416.13005}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D. Popescu, "General Néron desingularization" ''Nagoya Math. J.'' , '''100''' (1985) pp. 97–126 {{MR|0818160}} {{ZBL|0561.14008}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D. Popescu, "General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''104''' (1986) pp. 85–115 {{MR|0868439}} {{ZBL|0592.14014}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> D. Popescu, "Polynomial rings and their projective modules" ''Nagoya Math. J.'' , '''113''' (1989) pp. 121–128 {{MR|0986438}} {{ZBL|0663.13006}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> D. Popescu, "Letter to the Editor: General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''118''' (1990) pp. 45–53 {{MR|1060701}} {{ZBL|0685.14009}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> C. Rotthaus, "Rings with approximation property" ''Math. Ann.'' , '''287''' (1990) pp. 455–466 {{MR|1060686}} {{ZBL|0702.13007}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> M. Spivakovsky, "Non-existence of the Artin function for Henselian pairs" ''Math. Ann.'' , '''299''' (1994) pp. 727–729 {{MR|1286894}} {{ZBL|0803.13005}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> M. Spivakovsky, "A new proof of D. Popescu's theorem on smoothing of ring homomorphisms" ''J. Amer. Math. Soc.'' , '''294''' (to appear) {{MR|1647069}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> R. Swan, "Néron–Popescu desingularization" , ''Proc. Internat. Conf. Algebra and Geometry, Taipei, Taiwan, 1995'' , Internat. Press Boston (1998) {{MR|}} {{ZBL|0954.13003}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> B. Teissier, "Résultats récents sur l'approximation des morphisms en algèbre commutative,[d'après Artin, Popescu, André, Spivakovsky]" ''Sem. Bourbaki'' , '''784''' (1994) pp. 1–15</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> M. Van der Put, "A problem on coefficient fields and equations over local rings" ''Compositio Math.'' , '''30''' : 3 (1975) pp. 235–258 {{MR|}} {{ZBL|0304.13018}} </TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> J.J. Wavrik, "A theorem on solutions of analytic equations with applications to deformations of complex structures" ''Math. Ann.'' , '''216''' : 2 (1975) pp. 127–142 {{MR|0387649}} {{ZBL|0303.32018}} </TD></TR></table> |

## Revision as of 21:50, 30 March 2012

Let be a Noetherian local ring and its completion. has the Artin approximation property (in brief, has AP) if every finite system of polynomial equations over has a solution in if it has one in . In fact, has the Artin approximation property if and only if for every finite system of polynomial equations over the set of its solutions in is dense, with respect to the -adic topology, in the set of its solutions in . That is, for every solution of in and every positive integer there exists a solution of in such that modulo . The study of Artin approximation started with the famous papers of M. Artin [a3], [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 [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 [a24], a weaker result, namely that AP implies Henselian and universally Japanese, being proved in [a14], (5.4), and [a9]. The sufficiency gives a positive answer to Artin's conjecture [a5] and is a consequence (see [a21], (1.3), and [a27]) of the following theorem on general Néron desingularization ([a20], [a21], [a23], [a2], [a16], [a27], [a26]): A morphism between Noetherian rings is regular (i.e. it is flat and for every field that is a finite -algebra, the ring 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 is a regular morphism of Noetherian rings, then every finite system of polynomial equations over having a solution in can be enlarged to a finite system of polynomial equations over having a solution in , 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 . This is a partial positive answer to the Swan conjecture and, using [a15], proves the Bass–Quillen conjecture in the equicharacteristic case (see also [a22], [a27]).

Let be a Noetherian local ring. has the strong Artin approximation property (in brief, has SAP) if for every finite system of equations in over there exists a mapping with the following property: If satisfies modulo , , then there exists a solution of with modulo .

M. Greenberg [a13] proved that excellent Henselian discrete valuation rings have the strong Artin approximation property and M. Artin [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 has the strong Artin approximation property. In particular, has AP if and only if it has SAP. A special case of this is stated in [a11], together with many other applications.

When contains a field, some weaker results were stated in [a29], [a30]. In the above form, the result appeared in [a17], but the proof there has a gap in the non-separable case, which was repaired in [a14], Chap. 2. In [a8] it was noted that SAP is more easily handled using ultraproducts. Let be a non-principal ultrafilter on (i.e. an ultrafilter containing the filter of cofinite sets of ). The ultraproduct of with respect to is the factor of by the ideal of all such that the set . Assigning to the constant sequence one obtains a ring morphism . Using these concepts, easier proofs of the assertion were given in [a19] and [a10]. The easiest one is given in [a21], (4.5), where it is noted that the separation of in the -adic topology is Noetherian, that the canonical mapping is regular if is excellent and that is SAP if and only if for every finite system of polynomial equations over , for every positive integer and every solution of in , there exists a solution of in which lifts modulo . The result follows on applying general Néron desingularization to and using the implicit function theorem.

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

All these approximation properties were studied also for couples , were is not necessarily local and is not necessarily maximal. A similar proof shows that the Artin approximation property holds for a Henselian couple if is excellent [a21], (1.3). If is not Artinian, then 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 [a25].

A special type of Artin approximation theory was required in singularity theory. Such types were studied in [a14], Chaps. 3, 4. However, the result holds even in the following extended form: Let be an excellent Henselian local ring, its completion, the Henselization of , , in , a finite system of polynomial equations over and a formal solution of such that , , for some positive integers . Then there exists a solution of in such that , , and modulo , , for .

The proof is given in [a21], (3.6), (3.7), using ideas of H. Kurke and Pfister, who noticed that this assertion holds if has AP, where is an excellent Henselian local ring. If the sets of variables of are not "nested" (i.e. they are not totally ordered by inclusion), then the assertion does not hold, see [a7]. If is the convergent power series ring over and the algebraic power series rings are replaced by , then the theorem does not hold, see [a12]. Extensions of this theorem are given in [a28], [a27].

#### References

[a1] | M. André, "Artin's theorem on the solution of analytic equations in positive characteristic" Manuscripta Math. , 15 (1975) pp. 314–348 MR379493 |

[a2] | M. André, "Cinq exposés sur la desingularization" École Polytechn. Féd. Lausanne (1991) (Handwritten manuscript) |

[a3] | M. Artin, "On the solution of analytic equations" Invent. Math. , 5 (1968) pp. 277–291 MR0232018 |

[a4] | M. Artin, "Algebraic approximation of structures over complete local rings" Publ. Math. IHES , 36 (1969) pp. 23–58 MR0268188 Zbl 0181.48802 |

[a5] | M. Artin, "Construction techniques for algebraic spaces" , Actes Congres Internat. Math. , 1 (1970) pp. 419–423 MR0427316 Zbl 0232.14003 |

[a6] | M. Artin, "Versal Deformations and Algebraic Stacks" Invent. Math. , 27 (1974) pp. 165–189 MR0399094 Zbl 0317.14001 |

[a7] | J. Becker, "A counterexample to Artin approximation with respect to subrings" Math. Ann. , 230 (1977) pp. 195–196 MR0480508 Zbl 0359.13007 |

[a8] | J. Becker, J. Denef, L. Lipshitz, L. van den Dries, "Ultraproducts and approximation in local rings I" Invent. Math. , 51 (1979) pp. 189–203 MR0528023 Zbl 0416.13004 |

[a9] | M. Cipu, D. Popescu, "Some extensions of Néron's -desingularization and approximation" Rev. Roum. Math. Pures Appl. , 24 : 10 (1981) pp. 1299–1304 |

[a10] | J. Denef, L. Lipshitz, "Ultraproducts and approximation in local rings II" Math. Ann. , 253 (1980) pp. 1–28 MR0594530 Zbl 0426.13010 |

[a11] | R. Elkik, "Solutions d'equations à coefficients dans une anneau (!!) henselien" Ann. Sci. Ecole Norm. Sup. 4 , 6 (1973) pp. 533–604 MR345966 |

[a12] | A.M. Gabrielov, "The formal relations between analytic functions" Funkts. Anal. Prilozh. , 5 : 4 (1971) pp. 64–65 (In Russian) MR0302930 |

[a13] | M. Greenberg, "Rational points in Henselian discrete valuation rings" Publ. Math. IHES , 31 (1966) pp. 59–64 MR0207700 MR0191897 Zbl 0146.42201 Zbl 0142.00901 |

[a14] | H. Kurke, T. Mostowski, G. Pfister, D. Popescu, M. Roczen, "Die Approximationseigenschaft lokaler Ringe" , Lecture Notes Math. , 634 , Springer (1978) (Note: The proof of (3.1.1) is wrong) MR0485851 Zbl 0401.13013 |

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**How to Cite This Entry:**

Artin approximation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Artin_approximation&oldid=14797