Difference between revisions of "Fatou ring"
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| − | An [[Integral domain|integral domain]] $A$ with quotient field $K$ such that if each rational function $R\in K(X)$ that has a Taylor expansion at $0$ with coefficients in $A$, has a unitary and irreducible representation with coefficients in $A$; that is, for each $R\in K(X)\cap A[[X]]$ there are $P,Q\in A[X]$ such that $R=P/Q$, $Q(0)=1$ and $P$ and $Q$ are relatively prime in $K[X]$. Fatou's lemma | + | {{TEX|done}} |
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| + | An [[Integral domain|integral domain]] $A$ with quotient field $K$ such that if each rational function $R\in K(X)$ that has a Taylor expansion at $0$ with coefficients in $A$, has a unitary and irreducible representation with coefficients in $A$; that is, for each $R\in K(X)\cap A[[X]]$ there are $P,Q\in A[X]$ such that $R=P/Q$, $Q(0)=1$ and $P$ and $Q$ are relatively prime in $K[X]$. Fatou's lemma {{Cite|Fa}} states that $\mathbb{Z}$ is a Fatou ring. | ||
Equivalently, $A$ is a Fatou ring means that if a sequence $\{ a_n\}_{n\in\mathbb{N}}$ of elements of $A$ satisfies a linear recursion formula | Equivalently, $A$ is a Fatou ring means that if a sequence $\{ a_n\}_{n\in\mathbb{N}}$ of elements of $A$ satisfies a linear recursion formula | ||
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$$a_{n+s}+q_1a_{n+1}+\cdots+q_sa_n=0\textrm{ for }n\geq 0,$$ | $$a_{n+s}+q_1a_{n+1}+\cdots+q_sa_n=0\textrm{ for }n\geq 0,$$ | ||
| − | where $q_1, | + | where $q_1,\ldots,q_s\in K$ and $s$ is as small as possible, then $q_1,\ldots,q_s\in A$. |
If $A$ is a Fatou ring, then its quotient field $K$ is a [[Fatou extension|Fatou extension]] of $A$, but the converse does not hold. This is the reason why Fatou rings are sometimes called strong Fatou rings, while the domains $A$ such that $K$ is a Fatou extension of $A$ are called weak Fatou rings (in this latter case, every $R\in K(X)\cap A[[X]]$ has a unitary (not necessarily irreducible) representation with coefficients in $A$). | If $A$ is a Fatou ring, then its quotient field $K$ is a [[Fatou extension|Fatou extension]] of $A$, but the converse does not hold. This is the reason why Fatou rings are sometimes called strong Fatou rings, while the domains $A$ such that $K$ is a Fatou extension of $A$ are called weak Fatou rings (in this latter case, every $R\in K(X)\cap A[[X]]$ has a unitary (not necessarily irreducible) representation with coefficients in $A$). | ||
| − | The coefficients of the unitary and irreducible representation of every element of $K(X)\cap A((X))$ are almost integral over $A$ (see [[Fatou extension|Fatou extension]]). An integral domain $A$ is a Fatou ring if and only if every element of $K$ which is almost integral over $A$ belongs to $A$ | + | The coefficients of the unitary and irreducible representation of every element of $K(X)\cap A((X))$ are almost integral over $A$ (see [[Fatou extension|Fatou extension]]). An integral domain $A$ is a Fatou ring if and only if every element of $K$ which is almost integral over $A$ belongs to $A$ {{Cite|Ei}}; such domains are said to be completely integrally closed. For instance, a Noetherian domain (cf. also [[Noetherian ring|Noetherian ring]]) is completely integrally closed if and only if it is integrally closed (cf. also [[Integral domain|Integral domain]]). The rings of integers of number fields are completely integrally closed, and hence, Fatou rings. |
The notion may be extended by considering [[Formal power series|formal power series]] in non-commuting variables. The characterization of this generalized property is still (1998) an open question. | The notion may be extended by considering [[Formal power series|formal power series]] in non-commuting variables. The characterization of this generalized property is still (1998) an open question. | ||
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====References==== | ====References==== | ||
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| + | * {{Ref|Ei}} S. Eilenberg, "Automata, languages and machines", '''A''', Acad. Press (1974) | ||
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| + | * {{Ref|Fa}} P. Fatou, "Sur les séries entières à coefficients entiers" ''C.R. Acad. Sci. Paris Ser. A'', '''138''' (1904) pp. 342–344 {{ZBL|35.0257.02}} | ||
Latest revision as of 17:51, 28 May 2024
An integral domain $A$ with quotient field $K$ such that if each rational function $R\in K(X)$ that has a Taylor expansion at $0$ with coefficients in $A$, has a unitary and irreducible representation with coefficients in $A$; that is, for each $R\in K(X)\cap A[[X]]$ there are $P,Q\in A[X]$ such that $R=P/Q$, $Q(0)=1$ and $P$ and $Q$ are relatively prime in $K[X]$. Fatou's lemma [Fa] states that $\mathbb{Z}$ is a Fatou ring.
Equivalently, $A$ is a Fatou ring means that if a sequence $\{ a_n\}_{n\in\mathbb{N}}$ of elements of $A$ satisfies a linear recursion formula
$$a_{n+s}+q_1a_{n+1}+\cdots+q_sa_n=0\textrm{ for }n\geq 0,$$
where $q_1,\ldots,q_s\in K$ and $s$ is as small as possible, then $q_1,\ldots,q_s\in A$.
If $A$ is a Fatou ring, then its quotient field $K$ is a Fatou extension of $A$, but the converse does not hold. This is the reason why Fatou rings are sometimes called strong Fatou rings, while the domains $A$ such that $K$ is a Fatou extension of $A$ are called weak Fatou rings (in this latter case, every $R\in K(X)\cap A[[X]]$ has a unitary (not necessarily irreducible) representation with coefficients in $A$).
The coefficients of the unitary and irreducible representation of every element of $K(X)\cap A((X))$ are almost integral over $A$ (see Fatou extension). An integral domain $A$ is a Fatou ring if and only if every element of $K$ which is almost integral over $A$ belongs to $A$ [Ei]; such domains are said to be completely integrally closed. For instance, a Noetherian domain (cf. also Noetherian ring) is completely integrally closed if and only if it is integrally closed (cf. also Integral domain). The rings of integers of number fields are completely integrally closed, and hence, Fatou rings.
The notion may be extended by considering formal power series in non-commuting variables. The characterization of this generalized property is still (1998) an open question.
References
- [Ei] S. Eilenberg, "Automata, languages and machines", A, Acad. Press (1974)
- [Fa] P. Fatou, "Sur les séries entières à coefficients entiers" C.R. Acad. Sci. Paris Ser. A, 138 (1904) pp. 342–344 Zbl 35.0257.02
Fatou ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_ring&oldid=24794