Difference between revisions of "Formal power series"
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− | ''over a ring | + | ''over a ring $A$ in commuting variables $T_1,\ldots,T_N$'' |
An algebraic expression of the form | An algebraic expression of the form | ||
+ | $$ | ||
+ | F = \sum_{k=0}^\infty F_k | ||
+ | $$ | ||
− | + | where $F_k$ is a form of degree $k$ in $T_1,\ldots,T_N$ with coefficients in $A$. The minimal value of $k$ for which $F_k \ne 0$ is called the order of the series $F$, and the form $F_k$ is called the initial form of the series. | |
− | |||
− | where | ||
If | If | ||
− | + | $$ | |
− | + | F = \sum_{k=0}^\infty F_k \ \ \text{and}\ \ G = \sum_{k=0}^\infty G_k | |
− | + | $$ | |
are two formal power series, then, by definition, | are two formal power series, then, by definition, | ||
− | + | $$ | |
− | + | F + G = \sum_{k=0}^\infty F_k + G_k | |
− | + | $$ | |
and | and | ||
− | + | $$ | |
− | + | F \cdot G = \sum_{k=0}^\infty H_k | |
− | + | $$ | |
where | where | ||
+ | $$ | ||
+ | H_k = \sum_{j=0}^k F_j G_{k-j} \ . | ||
+ | $$ | ||
− | + | The set $A[[T_1,\ldots,T_N]]$ of all formal power series forms a ring under these operations. | |
− | + | A polynomial $F = \sum_{k=0}^n F_k$, where $F_k$ is a form of degree $k$, is identified with the formal power series $C = \sum_{k=0}^\infty C_k$ , where $C_k = F_k$ for $k \le n$ and $C_k = 0$ for $k > n$. This defines an imbedding $i$ of the polynomial ring $A[T_1,\ldots,T_N]$ into $A[[T_1,\ldots,T_N]]$. There is a topology defined on $A[[T_1,\ldots,T_N]]$ for which the ideals | |
+ | $$ | ||
+ | I_n = \{ F = \sum_{k=0}^\infty F_k \ :\ F_k = 0 \ \text{for}\ k \le n \} | ||
+ | $$ | ||
+ | form a fundamental system of neighbourhoods of zero. This topology is separable, the ring $A[[T_1,\ldots,T_N]]$ is complete relative to it, and the image of $A[T_1,\ldots,T_N]$ under the imbedding $i$ is everywhere dense in $A[[T_1,\ldots,T_N]]$. Relative to this topology, a power series $F = \sum_{k=0}^\infty F_k$ is the limit of its partial sums $F = \sum_{k=0}^n F_k$. | ||
− | A | + | Suppose that $A$ is a commutative ring with an identity. Then so is $A[[T_1,\ldots,T_N]]$. If $A$ is an integral domain, then so is $A[[T_1,\ldots,T_N]]$. A formal power series $F = \sum_{k=0}^\infty F_k$ is invertible in $A[[T_1,\ldots,T_N]]$ if and only if $F_0$ is invertible in $A$. If $A$ is Noetherian, then so is $A[[T_1,\ldots,T_N]]$. If $A$ is a local ring with maximal ideal $\mathfrak{m}$, then $A[[T_1,\ldots,T_N]]$ is a local ring with maximal ideal $\left\langle \mathfrak{m}, T_1,\ldots,T_N \right\rangle$ . |
− | + | If a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087050.png" /> is separable and complete in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087051.png" />-adic topology, then the [[Weierstrass preparation theorem]] is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087052.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087053.png" /> be a formal power series such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087054.png" /> the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087055.png" /> contains a term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087057.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087058.png" /> be the minimal index with this property. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087060.png" /> is an invertible formal power series and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087061.png" /> is a polynomial of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087062.png" />, where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087063.png" /> belong to the maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087064.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087066.png" /> are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087067.png" />. | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | If a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087050.png" /> is separable and complete in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087051.png" />-adic topology, then the Weierstrass preparation theorem is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087052.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087053.png" /> be a formal power series such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087054.png" /> the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087055.png" /> contains a term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087057.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087058.png" /> be the minimal index with this property. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087060.png" /> is an invertible formal power series and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087061.png" /> is a polynomial of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087062.png" />, where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087063.png" /> belong to the maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087064.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087066.png" /> are uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040870/f04087067.png" />. | ||
The ring of formal power series over a field or a discretely-normed ring is factorial. | The ring of formal power series over a field or a discretely-normed ring is factorial. | ||
Line 40: | Line 42: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , v. Nostrand (1960)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR> | ||
+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , v. Nostrand (1960)</TD></TR> | ||
+ | </table> | ||
Line 60: | Line 65: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Berstel (ed.) , ''Series formelles en variables noncommutatives et aplications'' , Lab. Inform. Théor. Programmation (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Nagata, "Local rings" , Interscience (1960)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Berstel (ed.) , ''Series formelles en variables noncommutatives et aplications'' , Lab. Inform. Théor. Programmation (1978)</TD></TR> | ||
+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Nagata, "Local rings" , Interscience (1960)</TD></TR> | ||
+ | </table> | ||
+ | |||
+ | {{TEX|part}} |
Revision as of 20:04, 6 December 2015
over a ring $A$ in commuting variables $T_1,\ldots,T_N$
An algebraic expression of the form $$ F = \sum_{k=0}^\infty F_k $$
where $F_k$ is a form of degree $k$ in $T_1,\ldots,T_N$ with coefficients in $A$. The minimal value of $k$ for which $F_k \ne 0$ is called the order of the series $F$, and the form $F_k$ is called the initial form of the series.
If $$ F = \sum_{k=0}^\infty F_k \ \ \text{and}\ \ G = \sum_{k=0}^\infty G_k $$ are two formal power series, then, by definition, $$ F + G = \sum_{k=0}^\infty F_k + G_k $$ and $$ F \cdot G = \sum_{k=0}^\infty H_k $$ where $$ H_k = \sum_{j=0}^k F_j G_{k-j} \ . $$
The set $A[[T_1,\ldots,T_N]]$ of all formal power series forms a ring under these operations.
A polynomial $F = \sum_{k=0}^n F_k$, where $F_k$ is a form of degree $k$, is identified with the formal power series $C = \sum_{k=0}^\infty C_k$ , where $C_k = F_k$ for $k \le n$ and $C_k = 0$ for $k > n$. This defines an imbedding $i$ of the polynomial ring $A[T_1,\ldots,T_N]$ into $A[[T_1,\ldots,T_N]]$. There is a topology defined on $A[[T_1,\ldots,T_N]]$ for which the ideals $$ I_n = \{ F = \sum_{k=0}^\infty F_k \ :\ F_k = 0 \ \text{for}\ k \le n \} $$ form a fundamental system of neighbourhoods of zero. This topology is separable, the ring $A[[T_1,\ldots,T_N]]$ is complete relative to it, and the image of $A[T_1,\ldots,T_N]$ under the imbedding $i$ is everywhere dense in $A[[T_1,\ldots,T_N]]$. Relative to this topology, a power series $F = \sum_{k=0}^\infty F_k$ is the limit of its partial sums $F = \sum_{k=0}^n F_k$.
Suppose that $A$ is a commutative ring with an identity. Then so is $A[[T_1,\ldots,T_N]]$. If $A$ is an integral domain, then so is $A[[T_1,\ldots,T_N]]$. A formal power series $F = \sum_{k=0}^\infty F_k$ is invertible in $A[[T_1,\ldots,T_N]]$ if and only if $F_0$ is invertible in $A$. If $A$ is Noetherian, then so is $A[[T_1,\ldots,T_N]]$. If $A$ is a local ring with maximal ideal $\mathfrak{m}$, then $A[[T_1,\ldots,T_N]]$ is a local ring with maximal ideal $\left\langle \mathfrak{m}, T_1,\ldots,T_N \right\rangle$ .
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
![]() |
where the range over
,
, has a well-defined meaning in
(as the unique limit of the finite sums
![]() |
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