Formal power series

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 $A$ is separable and complete in the $\mathfrak{m}$-adic topology, then the Weierstrass preparation theorem is true in $A[[T_1,\ldots,T_N]]$. Let $F$ be a formal power series such that for some $k$ the form $F_k$ contains a term $a T^k$, where $a \notin \mathfrak{m}$, and let $k$ be the minimal index with this property. Then $F = UP$, where $U$ is an invertible formal power series and $P$ is a polynomial of the form $T^k + a_{k-1}T^{k-1} + \cdots + a_0$, where the coefficients $a_i$ belong to the maximal ideal of $A[[T_1,\ldots,T_N]]$. The elements $U$ and $P$ are uniquely determined by $F$.

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 $A'$ be a ring containing $A$ (or provided with a ring homomorphism $\phi : A \rightarrow A'$), let $\mathfrak{a}'$ be an ideal in $A'$ and suppose that $A'$ is complete in the $\mathfrak{a}'$-adic topology on $A'$. Let $x_1,\ldots,x_n$ be elements of $\mathfrak{a}'$. Then an expression $$\sum_{ i_1,\ldots,i_n = 0 }^\infty c_{ i_1,\ldots,i_n } x_1^{i_1}\cdots x_n^{i_n}$$

where the $i_j$ range over $\mathbf{N} \cup \{0\} = \{0,1,2,\ldots \}$, $c_{ i_1,\ldots,i_n } \in A$, has a well-defined meaning in $A'$ (as the unique limit of the finite sums $$\sum_{ i_1,\ldots,i_n = 0 }^m c_{ i_1,\ldots,i_n } x_1^{i_1}\cdots x_n^{i_n}$$

as $m \rightarrow \infty$). Such an expression is also called a formal power series over $A$. Mapping $T_i$ to $x_i$, $i=1,\ldots,n$, defines a (continuous) homomorphism $A[[T_1,\ldots,T_n]] \rightarrow A'$. If this homomorphism is injective, the $x_1,\ldots,x_n$ are said to be analytically independent over $A$.

Let now $A$ be a field with a multiplicative norm on it (i.e. $\Vert ab \Vert = \Vert a \Vert\cdot\Vert b \Vert$), e.g. $A = \mathbf{C}$ with the usual norm or $A = \mathbf{Q}_p$, the rational field, with the norm $\Vert a \Vert = p^r$ if $r = -\nu_p(a)$, where $\nu_p$ is the $p$-adic valuation on $\mathbf{Q}$ ($\nu_p(m)$ for $m \in \mathbf{Z}$ is the exponent of the largest power of the prime number $p$ that divides $m$; $\nu_p(m/n) = \nu_p(m) - \nu_p(n)$). Now consider all formal power series $\sum c_{ i_1,\ldots,i_n } x_1^{i_1}\cdots x_n^{i_n}$ over $A$ such that there exist positive numbers $r_1,\ldots,r_n$ and $C$ such that $\Vert c_{ i_1,\ldots,i_n } \Vert \le C r_1^{i_1}\cdots r_n^{i_n}$. These form a subring of $A[[T_1,\ldots,T_n]]$, called the ring of convergent power series over $A$ and denoted by $A\{T_1,\ldots,T_n\}$ (or $A \langle\langle T_1,\ldots,T_n \rangle\rangle$, but the latter notation also occurs for the ring of power series in non-commuting variables over $A$). The Weierstrass preparation theorem also holds in $A\{T_1,\ldots,T_n\}$.

References

[a1]	J. Berstel (ed.) , Séries formelles en variables noncommutatives et applications , Lab. Inform. Théor. Programmation (1978)
[a2]	M. Nagata, "Local rings" , Interscience (1960)