Artin approximation theorem


Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field "k" are well-approximated by the algebraic functions on "k".

tatement of the theorem

Let

:x = "x"1, …, "x""n"

denote a collection of "n" indeterminates,

"k"x the ring of formal power series with indeterminates x over a field "k", and

: y = "y"1, …, "y""m"

a different set of indeterminates. Let

:"f"(x, y) = 0

be a system of polynomial equations in "k" [x, y] , and "c" a positive integer. Then given a formal power series solution ŷ(x) ∈ "k"x there is an algebraic solution y(x) consisting of algebraic functions such that

:ŷ(x) ≡ y(x) mod (x)"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 schemes.

References

*Artin, Michael. "Algebraic Spaces". Yale University Press, 1971.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Michael Artin — (photo de George Bergman). Naissance 1934 Hambourg (   …   Wikipédia en Français

  • Michael Artin — (photo by George Bergman) Born 1934 (age 76–77) Hambu …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… …   Wikipedia

  • Moduli space — In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as… …   Wikipedia

  • Formal moduli — In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the… …   Wikipedia

  • algebra — /al jeuh breuh/, n. 1. the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. 2. any of… …   Universalium

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… …   Wikipedia