GNU Free Documentation License, Version 1.2

- Examples
- User documentation
- Maintainer documentation
- Bugs, shortcomings and other ideas
- Main changes

In CoCoALib `factorization`

is a template class for representing
(partial) factorizations. Conceptually it comprises a **list of factors**
and **their multiplicities**, and an extra **remaining factor** (which may be,
for instance, an unfactorized part, or an invertible element).

The class itself imposes few conditions: the factors in the list
cannot be invertible or zero-divisors, and their multiplicities are
all positive. The remaining factor is a non-zero-divisor. The exact
characteristics of the factors depend on the function which generated
the `factorization`

. Naturally, the vectors returned by
`myFactors`

and `myMultiplicities`

will be of the same length.
In particular, an **empty factorization is permitted** (*i.e.* with empty
lists of factors and multiplicities).

See also: the functions `factor`

, `SqFreeFactor`

, `ContentFreeFactor`

(in section `factor`

),
and the functions `factor`

(in section `NumTheory`

)

`factorization(RemFactor)`

specifies an initial remaining factor, the factor/multiplicity lists are empty`factorization(facs, mults, RemFactor)`

specifies initial values for the 3 components

Let `FactorInfo`

be of type `factorization<T>`

. These are the accessor functions:

`FactorInfo.myFactors()`

all the factors as a`std::vector<T>`

(read-only)`FactorInfo.myMultiplicities()`

all the multiplicities as a`std::vector<long>`

(read-only)`FactorInfo.myRemainingFactor()`

the remaining factor (read-only)

For **better readability of code** using `factorization`

we recommend using const ref
aliases for the lists of factors and multiplicities; for instance

const factorization<RingElem> FactorInfo = factor(f); const vector<RingElem>& facs = FactorInfo.myFactors(); const vector<long>& mults = FactorInfo.myMultiplicities(); ... // code using the arrays "facs" and "mults"

Let `FactorInfo`

be of type `factorization<T>`

. These are the operations available:

`FactorInfo.myAppend(fac, mult)`

appends a new factor with its multiplicity`FactorInfo.myNewRemainingFactor(RemFactor)`

sets`RemFactor`

as the remaining factor

Being template code it's all in the header file. It's mostly fairly straightfoward.

The main point to note is that `ourCheckNotZeroDiv`

and `ourCheckNotUnit`

need to be written by hand for each instantiation -- this is enforced by the
absence of a default template impl. **Note that** the impls for `DUPFp`

are
defined in the file `DUPFp.H`

.

The fn `ourCheckCompatibility`

is needed for `RingElem`

but not for other
types (so the default impl is empty). It simply checks that all the factors
belong to the same ring (equiv. that they belong to ring of `myRemainingFactorValue`

).

In CoCoALib there are just 4 instantiations of this template:

`factorization<BigInt>`

for the fns`factor`

&`factor_TrialDiv`

&`factor_PollardRho`

in`NumTheory`

`factorization<RingElem>`

for the fns`factor`

and`SqFreeFactor`

and`ContentFreeFactor`

in`PolyRing`

(actually`TmpFactor`

)`factorization<long>`

`factorization<DUPFp>`

It would be safer to have pairs of factor-and-multiplicity rather than two separate vectors whose length must be the same. However it may be less convenient for the user.

Maybe add fn to get length of a `factorization`

? (same as length of `myFactors()`

)

Maybe add fn to get ring of a `factorization<RingElem>`

?

Maybe add fn to change the multiplicity of a factor?

Bruns questioned the necessity of the restriction that factors be non-zero-divisiors and non-units. JAA prefers to apply these restrictions for the time being, because it will be easier to relax the restrictions later than it would be to tighten them (might break some existing code).

Bruns/Soeger asked whether requiring all factors to be in the same ring is necessary (esp. once CoCoA allows arithmetic between different rings). They cite the example of factors in ZZ[x] and remaining factor in QQ.

**2014**

- March: fields are now private: new accessor fns; new append fn, and new fn to update rem factor

**2012**

- October: chose "myMultiplicities" rather than "myExponents" as the field name.
- April: first version of doc (v0.9950)