mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Artin approximation theorem is a fundamental result of in

deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...

which implies that

formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...

with coefficients in a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'' are well-approximated by the

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additio ...

s on ''k''. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case

k = \Complex

); and an algebraic version of this theorem in 1969.

Statement of the theorem

Let

\mathbf = x_1, \dots,  x_n

denote a collection of ''n'' indeterminates,

k \mathbf

the

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

of formal power series with indeterminates

\mathbf

over a field ''k'', and

\mathbf = y_1, \dots,  y_n

a different set of indeterminates. Let :

f(\mathbf, \mathbf) = 0

be a system of

polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...

s in

k mathbf, \mathbf /math>, and ''c'' a positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. Then given a formal power series solution

\hat(\mathbf) \in k \mathbf

, there is an algebraic solution

\mathbf(\mathbf)

consisting of

s (more precisely, algebraic power series) such that :

\hat(\mathbf)  \equiv \mathbf(\mathbf) \bmod (\mathbf)^c.

Discussion

Given any desired positive integer ''c'', this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by ''c''. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as

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s. See also:

Artin's criterion In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these condi ...

Alternative statement

The following alternative statement is given in Theorem 1.12 of . Let

R

be a field or an excellent discrete valuation ring, let

A

be the

henselization In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...

of an

R

-algebra of finite type at a prime ideal, let ''m'' be a proper ideal of

A

, let

\hat

be the ''m''-adic completion of

A

, and let :

F\colon (A\text) \to (\text),

be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then for any integer ''c'' and any

\overline \in F(\hat)

, there is a

\xi \in F(A)

such that :

\overline \equiv \xi \bmod m^c

References

* * *{{citation, last=Raynaud, first= Michel, author-link=Michel Raynaud, title=Travaux récents de M. Artin, journal=

Séminaire Nicolas Bourbaki The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. It is one of the major institutions of contemporary mathematics, and a baro ...

, volume= 11 , year=1971, issue=363, pages= 279-295, url=http://www.numdam.org/book-part/SB_1968-1969__11__279_0/, mr=3077132 Moduli theory Commutative algebra Theorems about algebras