View source for Fatou extension

''of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120020/f1200202.png" />''

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" />.

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.

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:

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" />;

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]]);

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,

<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>

(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" />).

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:

<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>

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" />.

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:

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" />.

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" />.

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" />.

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]]].

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====
<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>