# 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.

How to Cite This Entry:
Formal power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_power_series&oldid=36864
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article