Fatou extension

of a commutative ring $A$

A commutative ring $B$ containing $A$ such that each formal power series $\alpha \in A [ [ X ] ]$ which is $B$-rational is in fact $A$-rational. Recall that a formal power series $\alpha$ is $R$-rational, $R$ a commutative ring, if there exist two polynomials $P , Q \in R [ X ]$ such that $Q ( 0 ) = 1$ and $\alpha = P / Q$, that is, $\alpha$ is equal to the formal expansion of $P \sum _ { n = 0 } ^ { \infty } ( Q - 1 ) ^ { n }$. For instance, if $K \subseteq L$ is a field extension (cf. also Extension of a field), then $L$ is a Fatou extension of $K$.

Fatou extensions are well characterized in the integral case. Thus, from now on, $A$ is supposed to be an integral domain with quotient field $K$. The example above shows that an integral domain $B$ containing $A$ is a Fatou extension of $A$ if and only if the ring $B \cap K$ is a Fatou extension of $A$. If the integral domain $A$ is Noetherian (cf. Noetherian ring), then its quotient field $K$ is a Fatou extension of $A$, and, hence, every integral domain containing $A$ is a Fatou extension of $A$. Many rings are Noetherian: for instance, every finitely generated $\bf Z$-algebra is Noetherian.

For a rational function $R \in K ( X )$, there are several representations of the form $R = P / Q$ with $P , Q \in K [ X ]$. Such a representation is said to be:

a) unitary if the non-zero coefficient of $Q$ corresponding to the lowest degree is $1$;

b) irreducible if $P$ and $Q$ are relatively prime in $K [ X ]$ (cf. also Mutually-prime numbers);

c) with coefficients in $A$ if $P , Q \in A [ X ]$. Let $A ( X )$ denote the set of rational functions with a unitary representation with coefficients in $A$, and let $A ( ( X ) )$ denote the set of Laurent power series, that is,

\begin{equation*} A ( ( X ) ) = \{ \sum _ { n \geq n _ { 0 } } ^ { \infty } a _ { n } X ^ { n } : n _ { 0 } \in \mathbf{Z} , a _ { n } \in A \} \end{equation*}

(these notations extend the classical notations $K ( X )$ and $K ( ( X ) )$).

To say that the integral domain $B$ is a Fatou extension of $A$ is nothing else than to write:

\begin{equation*} B ( X ) \bigcap A ( ( X ) ) = A ( X ); \end{equation*}

in other words, each rational function $R \in L ( X )$, where $L$ denotes the quotient field of $B$, which has a unitary representation with coefficients in $B$ and a Laurent expansion at $0$ with coefficients in $A$, has a unitary representation with coefficients in $A$.

A rational function $R \in K ( X )$ has a unique unitary and irreducible representation. With respect to this representation, there are two main results:

1) The ring $A ( X )$ is the set of elements of $K ( X ) \cap A ( ( X ) )$ which admit a unitary and irreducible representation whose coefficients are integral over $A$.

2) For every element of $K ( X ) \cap A ( ( X ) )$, the coefficients of the unitary and irreducible representation are almost integral over $A$. Recall that an element $x$ of $K$ is almost integral over $A$ if there exists a non-zero element $d$ of $A$ such that $d x ^ { n }$ belongs to $A$ for each positive integer $n$. Each element of $K$ which is integral over $A$ is almost integral over $A$.

An integral domain $B$ containing $A$ is a Fatou extension of $A$ if and only if each element of $K$ which is both integral over $B$ and almost integral over $A$ is integral over $A$ [a1]. The Noetherian case considered above follows from the fact that if $A$ is Noetherian, then each element of $K$ which is almost integral over $A$ is integral over $A$.

The definition of Fatou extension may be easily extended to semi-ring extensions. Then, ${\bf Q}_ +$ is a Fatou extension of $\mathbf{N}$, while $\bf Z$ is not a Fatou extension of $\mathbf{N}$, nor $\mathbf{R} _ { + }$ of ${\bf Q}_ +$ [a2].

Moreover, the notion may be considered for formal power series in non-commuting variables, which have applications in system and control theory [a3]. It turns out that the previous characterization in the integral case still holds.

References

[a1]	P.-J. Cahen, J.-L. Chabert, "Eléments quasi-entiers et extensions de Fatou" J. Algebra , 36 (1975) pp. 185–192
[a2]	J. Berstel, C. Reutenauer, "Rational series and their languages" , Springer (1988)
[a3]	A. Salomaa, M. Soittola, "Automata-theoretic aspects of formal power series" , Springer (1978)