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''of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200202.png" />''
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A [[Commutative ring|commutative ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200203.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200204.png" /> such that each [[Formal power series|formal power series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200205.png" /> which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200206.png" />-rational is in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200207.png" />-rational. Recall that a formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200208.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002010.png" />-rational, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002011.png" /> a commutative ring, if there exist two polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002014.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002015.png" /> is equal to the formal expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002016.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002017.png" /> is a field extension (cf. also [[Extension of a field|Extension of a field]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002018.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002019.png" />.
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Fatou extensions are well characterized in the integral case. Thus, from now on, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002020.png" /> is supposed to be an [[Integral domain|integral domain]] with quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002021.png" />. The example above shows that an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002022.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002023.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002024.png" /> if and only if the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002025.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002026.png" />. If the integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002027.png" /> is Noetherian (cf. [[Noetherian ring|Noetherian ring]]), then its quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002028.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002029.png" />, and, hence, every integral domain containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002030.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002031.png" />. Many rings are Noetherian: for instance, every finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002032.png" />-algebra is Noetherian.
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''of a commutative ring $A$''
  
For a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002033.png" />, there are several representations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002035.png" />. Such a representation is said to be:
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A [[Commutative ring|commutative ring]] $B$ containing $A$ such that each [[Formal power series|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 $K$.
  
a) unitary if the non-zero coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002036.png" /> corresponding to the lowest degree is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002037.png" />;
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Fatou extensions are well characterized in the integral case. Thus, from now on, $A$ is supposed to be an [[Integral domain|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.
  
b) irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002039.png" /> are relatively prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002040.png" /> (cf. also [[Mutually-prime numbers|Mutually-prime numbers]]);
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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:
  
c) with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002041.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002043.png" /> denote the set of rational functions with a unitary representation with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002045.png" /> denote the set of Laurent power series, that is,
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a) unitary if the non-zero coefficient of $Q$ corresponding to the lowest degree is $1$;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002046.png" /></td> </tr></table>
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b) irreducible if $P$ and $Q$ are relatively prime in $K [ X ]$ (cf. also [[Mutually-prime numbers|Mutually-prime numbers]]);
  
(these notations extend the classical notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002048.png" />).
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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,
  
To say that the integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002049.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002050.png" /> is nothing else than to write:
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\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*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002051.png" /></td> </tr></table>
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(these notations extend the classical notations $K ( X )$ and $K ( ( X ) )$).
  
in other words, each rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002053.png" /> denotes the quotient field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002054.png" />, which has a unitary representation with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002055.png" /> and a Laurent expansion at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002056.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002057.png" />, has a unitary representation with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002058.png" />.
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To say that the integral domain $B$ is a Fatou extension of $A$ is nothing else than to write:
  
A rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002059.png" /> has a unique unitary and irreducible representation. With respect to this representation, there are two main results:
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\begin{equation*} B ( X ) \bigcap A ( ( X ) ) = A ( X ); \end{equation*}
  
1) The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002060.png" /> is the set of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002061.png" /> which admit a unitary and irreducible representation whose coefficients are integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002062.png" />.
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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 $A$.
  
2) For every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002063.png" />, the coefficients of the unitary and irreducible representation are almost integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002064.png" />. Recall that an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002066.png" /> is almost integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002067.png" /> if there exists a non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002070.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002071.png" /> for each positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002072.png" />. Each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002073.png" /> which is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002074.png" /> is almost integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002075.png" />.
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A rational function $R \in K ( X )$ has a unique unitary and irreducible representation. With respect to this representation, there are two main results:
  
An integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002076.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002077.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002078.png" /> if and only if each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002079.png" /> which is both integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002080.png" /> and almost integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002081.png" /> is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002082.png" /> [[#References|[a1]]]. The Noetherian case considered above follows from the fact that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002083.png" /> is Noetherian, then each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002084.png" /> which is almost integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002085.png" /> is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002086.png" />.
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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 $A$.
  
The definition of Fatou extension may be easily extended to semi-ring extensions. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002087.png" /> is a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002088.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002089.png" /> is not a Fatou extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002090.png" />, nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f12002092.png" /> [[#References|[a2]]].
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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 $A$.
  
Moreover, the notion may be considered for formal power series in non-commuting variables, which have applications in system and control theory [[#References|[a3]]]. It turns out that the previous characterization in the integral case still holds.
+
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 $A$ [[#References|[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 $A$.
 +
 
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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 ${\bf Q}_ +$ [[#References|[a2]]].
 +
 
 +
Moreover, the notion may be considered for [[formal power series]] in non-commuting variables, which have applications in system and control theory [[#References|[a3]]]. It turns out that the previous characterization in the integral case still holds.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.-J. Cahen,  J.-L. Chabert,  "Eléments quasi-entiers et extensions de Fatou"  ''J. Algebra'' , '''36'''  (1975)  pp. 185–192</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Berstel,  C. Reutenauer,  "Rational series and their languages" , Springer  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Salomaa,  M. Soittola,  "Automata-theoretic aspects of formal power series" , Springer  (1978)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  P.-J. Cahen,  J.-L. Chabert,  "Eléments quasi-entiers et extensions de Fatou"  ''J. Algebra'' , '''36'''  (1975)  pp. 185–192</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  J. Berstel,  C. Reutenauer,  "Rational series and their languages" , Springer  (1988)</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top">  A. Salomaa,  M. Soittola,  "Automata-theoretic aspects of formal power series" , Springer  (1978)</td></tr>
 +
</table>

Latest revision as of 11:51, 24 December 2020

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)
How to Cite This Entry:
Fatou extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_extension&oldid=18283
This article was adapted from an original article by Jean-Luc Chabert (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article