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
factorization. Naturally, the vectors returned by
myMultiplicities will be of the same length.
See also: the functions
ContentFreeFactor (in section
and the function
SmoothFactor (in section
factorization(RemFactor)specifies an initial remaining factor, the factor/multiplicity lists are empty
factorization(facs, mults, RemFactor)specifies initial values for the 3 components
FactorInfo be of type
factorization<T>. These are the accessor functions:
FactorInfo.myFactors()all the factors as a
FactorInfo.myMultiplicities()all the multiplicities as a
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"
FactorInfo be of type
factorization<T>. These are the operations available:
FactorInfo.myAppend(fac, mult)appends a new factor with its multiplicity
RemFactoras 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
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
defined in the file
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
In CoCoALib there are just 4 instantiations of this template:
factorization<BigInt>for the fns
factorization<RingElem>for the fns
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
Maybe add fn to get ring of a
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.