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
2012