# FractionField

© 2005,2012 John Abbott, Anna M. Bigatti
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

## User documentation for FractionField

A `FractionField` is an abstract class (inheriting from `ring`) representing a fraction field of an effective GCD domain.

See `RingElem` FractionField for operations on its elements.

### Pseudo-constructors

• `NewFractionField(R)` -- creates a new `ring`, more precisely a `FractionField`, whose elements are formal fractions of elements of `R` (where `R` is a true GCD domain, see `IsTrueGCDDomain` in `ring`).
• `RingQQ()` -- produces the CoCoA `ring` which represents QQ, the field of rational numbers, fraction field of `RingZZ`. Calling `RingQQ()` several times will always produce the same unique ring in CoCoALib.
• `FractionField(R)` -- sort of downcast the ring `R` to a fraction field; will throw an `ErrorInfo` object with code `ERR::NotFracField` if needed.

### Query and cast

Let `S` be a `ring`

• `IsFractionField(S)` -- `true` iff `S` is actually a `FractionField`
• `FractionFieldPtr(S)` -- pointer to the fraction field impl (for calling mem fns); will throw an `ErrorInfo` object with code `ERR::NotFracField` if needed.

### Operations on FractionField

In addition to the standard `ring` operations, a `FractionField` may be used in other functions.

Let `FrF` be a `FractionField` built as `NewFractionField(R)` with `R` a `ring`

• `BaseRing(FrF)` -- the `ring` it is the `FractionField` of -- an identical copy of `R`, not merely an isomorphic `ring`.

### Homomorphisms

• `EmbeddingHom(FrF)` -- `BaseRing(FrF)` --> `FrF`
• `InducedHom(FrF, phi)` -- phi: `BaseRing(K)` --> `codomain(phi)`

## Maintainer documentation for FractionField, FractionFieldBase, FractionFieldImpl

The class `FractionField` is wholly analogous to the class `ring`, i.e. a reference counting smart pointer. The only difference is that `FractionField` knows that the `myRingPtr` data member actually points to an instance of a class derived from `FractionFieldBase` (and so can safely apply a `static_cast` when the pointer value is accessed).

`FractionFieldBase` is an abstract class derived from `RingBase`. It adds a few pure virtual functions to those contained in `RingBase`:

```  virtual const ring& myBaseRing() const;
virtual ConstRawPtr myRawNum(ConstRawPtr rawq) const; // NB result belongs to BaseRing!!
virtual ConstRawPtr myRawDen(ConstRawPtr rawq) const; // NB result belongs to BaseRing!!
virtual const RingHom& myEmbeddingHom() const;
virtual RingHom myInducedHomCtor(const RingHom& phi) const;
```

`myBaseRing` returns a reference to the `ring` (guaranteed to be an effective GCD domain) supplied to the constructor.

`myRawNum` (resp. `myRawDen`) produces a raw pointer to a value belonging to `BaseRing` ( and *NOT* to the `FractionField`!); these two functions *practically* *oblige* the implementation of `FractionField` to represent a value as a pair of raw values "belonging" to the `BaseRing`. Note that, while the value belongs to `BaseRing`, the resources are owned by the `FractionField`!!

`EmbeddingHom` returns the embedding homomorphism from the `BaseRing` into the `FractionField`; it actually returns a reference to a fixed homomorpism held internally.

`InducedHom` creates a new homomorpism from the `FractionField` to another `ring` S given a homomorpism from the `BaseRing` to S.

`FractionFieldImpl` implements a general fraction field. Its elements are just pairs of `RawValue`s belonging to the `BaseRing` (see the struct `FractionFieldElem`). For this implementation the emphasis was clearly on simplicity over speed (at least partly because we do not expect `FractionFieldImpl` to be used much). For polynomials whose coefficients lie in a `FractionField` we plan to implement a specific `ring` which uses a common denominator representation for the whole polynomial. If you want to make this code faster, see some of the comments in the bugs section.

Important: while fractions are guaranteed to be reduced (i.e. no common factor exists between numerator and denominator), it is rather hard to ensure that they are canonical since in general we can multiply numerator and denominator by any unit. See a bug comment about normalizing units.

## Bugs, Shortcomings and other ideas

The functions `myNew` are not exception safe: memory would be leaked if space for the numerator were successfully allocated while allocation for the denominator failed -- nobody would clean up the resources consumed by the numerator. Probably need a sort of `auto_ptr` for holding temporary bits of a value.

Should partial homomorphisms be allowed: e.g. from QQ to ZZ/(3)? Mathematically it is a bit dodgy, but in practice all works out fine provided you don't divide by zero. I think it would be too useful (e.g. for chinese remaindering methods) to be banned. Given phi:ZZ->ZZ[x] it might be risky to induce QQ->ZZ[x]; note that ker(phi)=0, so this is not a sufficient criterion!

Currently you can make a `FractionField` only from a ring satisfying `IsTrueGCDDomain`; in principle one could create a `FractionFieldImpl` of any integral domain (it just wouldn't be possible to cancel factors without a GCD -- so probably not terribly practical). I'll wait until someone really needs it before allowing it.

It is not clear how to make the denominator positive when the GCD domain is ZZ (so the fraction field is QQ). In general we would need the GCD domain to supply a normalizing unit: such a function could return 1 unless we have some special desire to normalize the denominator in a particular way. HERE'S A CONUNDRUM: FractionField(Q[x]) -- all rationals are units, and so we could end up with horrible representations like (22/7)/(22/7) instead of just 1. MUST FIX THIS!!

The printing function is TERRIBLE!

FASTER + and -
Addition and subtraction can be done better: let h be the GCD of the two denominators, suppose we want to compute a/bh + c/dh (where gcd(a,bh) = gcd(c, dh) = gcd(b,d) = 1 i.e. h = gcd(B,D) where B,D are the denoms) If h = 1 then there is no cancellation, o/w gcd(ad+bc, bdh) = gcd(ad+bc, h), so we can use a simpler gcd computation to find the common factor.

FASTER * and /
Multiplication and division can also be made a little faster by simplifying the GCD computation a bit. The two cases are essentially the same, so I shall consider just multiplication. Assuming inputs are already reduced (i.e. there is no common factor between numerator and denominator). To compute (a/b)*(c/d), first calculate h1 = gcd(a, d) and h2 = gcd(b, c). The result is then: num = (a/h1)*(c/h2) & den = (b/h2)*(d/h1) and this is guaranteed to be in reduced form.

`myIsRational` is incomplete: it will fail to recognise rationals whose numerator and denominator have been multiplied by non-trivial units. BAD BUG! Ironically `myIsInteger` does work correctly.