Formal power series
over a ring in commuting variables
An algebraic expression of the form
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where is a form of degree
in
with coefficients in
. The minimal value of
for which
is called the order of the series
, and the form
is called the initial form of the series.
If
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are two formal power series, then, by definition,
![]() |
and
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where
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The set of all formal power series forms a ring under these operations.
A polynomial , where
is a form of degree
, is identified with the formal power series
, where
for
and
for
. This defines an imbedding
of the polynomial ring
into
. There is a topology defined on
for which the ideals
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form a fundamental system of neighbourhoods of zero. This topology is separable, the ring is complete relative to it, and the image of
under the imbedding
is everywhere dense in
. Relative to this topology, a power series
is the limit of its partial sums
.
Suppose that is a commutative ring with an identity. Then so is
. If
is an integral domain, then so is
. A formal power series
is invertible in
if and only if
is invertible in
. If
is Noetherian, then so is
. If
is a local ring with maximal ideal
, then
is a local ring with maximal ideal
.
If a local ring is separable and complete in the
-adic topology, then the Weierstrass preparation theorem is true in
. Let
be a formal power series such that for some
the form
contains a term
, where
, and let
be the minimal index with this property. Then
, where
is an invertible formal power series and
is a polynomial of the form
, where the coefficients
belong to the maximal ideal of
. The elements
and
are uniquely determined by
.
The ring of formal power series over a field or a discretely-normed ring is factorial.
Rings of formal power series in non-commuting variables have also been studied.
References
[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[2] | O. Zariski, P. Samuel, "Commutative algebra" , 2 , v. Nostrand (1960) |
Comments
Power series in non-commuting variables are becoming rapidly more important and find applications in combinatorics (enumerative graph theory), computer science (automata) and system and control theory (representation of the input-output behaviour of non-linear systems, especially bilinear systems); cf. the collection [a1] for a first idea.
Let be a ring containing
(or provided with a ring homomorphism
), let
be an ideal in
and suppose that
is complete in the
-adic topology on
. Let
be elements of
. Then an expression
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where the range over
,
, has a well-defined meaning in
(as the unique limit of the finite sums
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as ). Such an expression is also called a formal power series over
. Mapping
to
,
, defines a (continuous) homomorphism
. If this homomorphism is injective, the
are said to be analytically independent over
.
Let now be a field with a multiplicative norm on it (i.e.
), e.g.
with the usual norm or
, the rational field, with the norm
if
, where
is the
-adic valuation on
(
for
is the exponent of the largest power of the prime number
that divides
;
). Now consider all formal power series
over
such that there exists positive numbers
and
such that
. These form a subring of
, called the ring of convergent power series over
and denoted by
(or
, but the latter notation also occurs for the ring of power series in non-commuting variables over
). The Weierstrass preparation theorem also holds in
.
References
[a1] | J. Berstel (ed.) , Series formelles en variables noncommutatives et aplications , Lab. Inform. Théor. Programmation (1978) |
[a2] | M. Nagata, "Local rings" , Interscience (1960) |
Formal power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_power_series&oldid=36859