Artin approximation theorem
Encyclopedia
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 function
s on k.
denote a collection of n indeterminate
s,
kx
the ring
of formal power series with indeterminates x over a field k, and
a different set of indeterminates. Let
be a system of polynomial equations in k[x, y], and c a positive integer
. Then given a formal power series solution ŷ(x) ∈ kx
there is an algebraic solution y(x) consisting of algebraic function
s such that
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Artin approximation theorem is a fundamental result of Michael Artin
Michael Artin
Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...
in deformation theory
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 Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
which implies that formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
with coefficients in a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k are well-approximated by the algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s on k.
Statement of the theorem
Let- x = x1, …, xn
denote a collection of n indeterminate
Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that does not stand for anything else but itself. In particular it does not designate a constant, or a parameter of the problem, it is not an unknown that could be solved for, it is not a variable designating a...
s,
k
X
X is the twenty-fourth letter in the basic modern Latin alphabet.-Uses:In mathematics, x is commonly used as the name for an independent variable or unknown value. The usage of x to represent an independent or unknown variable can be traced back to the Arabic word šay شيء = “thing,” used in Arabic...
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of formal power series with indeterminates x over a field k, and
- y =
a different set of indeterminates. Let
be a system of polynomial equations in k[x, y], and c a positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. Then given a formal power series solution ŷ(x) ∈ k
X
X is the twenty-fourth letter in the basic modern Latin alphabet.-Uses:In mathematics, x is commonly used as the name for an independent variable or unknown value. The usage of x to represent an independent or unknown variable can be traced back to the Arabic word šay شيء = “thing,” used in Arabic...
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s 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 schemeScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s.