Artin Approximation Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 infinitesima ...
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 su ...
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 an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...
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 (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
of formal
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
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 (mathematics), field, often the field of the rational numbers. For example, x^5-3x+1=0 is a ...
s in k mathbf, \mathbf/math>, and ''c'' a positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. Then given a formal power series solution \hat(\mathbf) \in k \mathbf, there is an algebraic solution \mathbf(\mathbf) consisting of
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...
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
scheme Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'', a BBC Scotland documentary TV series * The Scheme (band), an English pop band * ''The Scheme'', an action role-playing video game for the PC-8801, made by Quest Corporation * ...
s. See also: Artin's criterion.


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 ...
at a prime ideal of an R-algebra of finite type, 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.


See also

* Ring with the approximation property *
Popescu's theorem In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and ...
* Artin's criterion


References

* * *{{citation, last=Raynaud, first= Michel, author-link=Michel Raynaud, title=Travaux récents de M. Artin, journal=
Séminaire Nicolas Bourbaki The (from French: 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 barometer of m ...
, 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