SparsePolyRing is an abstract class (inheriting from PolyRing)
representing rings of polynomials; in particular, rings of sparse
multivariate polynomials (i.e. written in a sparse representation)
with a special view towards computing Groebner bases and other related
operations. This means that the operations offered by a
SparsePolyRing on its own values are strongly oriented towards those
needed by Buchberger's algorithm.
A polynomial is viewed abstractly as a formal sum of ordered terms
(with default ordering StdDegRevLex). Each term is a formal
product of a non-zero coefficient (belonging to the coefficient ring),
and a power product of indeterminates (belonging to the PPMonoid of
the polynomial ring). The ordering is determined by the PPOrdering
on the power products: distinct terms in a polynomial have distinct power
products. The zero polynomial is conceptually the formal sum of no terms;
all other polynomials have a leading term being the one with the largest
power product (PPMonoidElem) in the given ordering.
See RingElem SparsePolyRing for operations on its elements.
Currently there are three functions to create a polynomial ring:
NewPolyRing(CoeffRing, IndetNames)CoeffRing
and having indeterminates whose names are given in IndetNames (which
is of type vector<symbol>). The PP ordering is StdDegRevLex (with
indet(j) > indet(j+1) for each j).
NewPolyRing(CoeffRing, IndetNames, ord)CoeffRing
and having indeterminates whose names are given in IndetNames (which
is of type vector<symbol>). The PP ordering is given by ord
(a PPOrdering or PPOrdering PPOrderingCtor).
NewPolyRing(CoeffRing, PPM)CoeffRing and
with power products in PPM which is a power product monoid which specifies
how many indeterminates, their names, and the ordering on them.
SparsePolyRing(R)R to a sparse poly ring;
will throw an ErrorInfo object with code ERR::NotSparsePolyRing if needed.
In place of NewPolyRing you may use NewPolyRing_DMPI; this creates a
sparse poly ring which uses a more compact internal representation (which probably
makes computations slightly faster), but it necessarily uses a PPMonoidOv for
the power products. There is also NewPolyRing_DMPII which uses a still more
compact internal representation, but which may be used only when the coefficients
are in a small finite field and the power products are in a PPMonoidOv.
Let R be an object of type ring.
IsSparsePolyRing(R) -- true if R is actually SparsePolyRing
SparsePolyRingPtr(R) -- pointer to impl of R (for calling mem fns);
will throw an ErrorInfo object with code ERR::NotSparsePolyRing if needed
In addition to the standard PolyRing operations, a
SparsePolyRing may be used in other functions.
Let P be an object of type SparsePolyRing.
PPM(P) -- the PPMonoid of P.
OrdMat(P) -- a matrix defining the term ordering on P.
GradingDim(P) -- the dimension of the grading on P (may be 0).
GradingMat(P) -- the matrix defining the grading on P
RandomLinearForm(P, N) -- produce a non-zero random linear form from P with coeffs at most N
A SparsePolyIter (class defined in SparsePolyRing.H) is a way to
iterate through the summands in the polynomial without knowing the
(private) details of the concrete implementation currently in use.
See also the functions coefficients, CoefficientsWRT,
CoeffVecWRT in RingElem.
Let f denote a non-const element of P.
Let it1 and it2 be two SparsePolyIters running over the same polynomial.
BeginIter(f) -- a SparsePolyIter pointing to the first term in f.
EndIter(f) -- a SparsePolyIter pointing to one-past-the-last term
in f.
Changing the value of f invalidates all iterators over f.
coeff(it1) -- read-only access to the coeff of the current term
PP(it1) -- read-only access to the pp of the current term
++it1 -- advance it1 to next term, return new value of it1
it1++ -- advance it1 to next term, return copy of old value of it1
it1 == it2 -- true iff it1 and it2 point to the same term;
throws CoCoA::ErrorInfo with code
ERR::MixedPolyIters
if it1 and it2 are over different polys.
it1 != it2 -- same as !(it1 == it2)
IsEnded(it1) -- true iff it1 is pointing at the one-past-the-last term
The exact nature of a term in a polynomial is hidden from public view: it is not possible to get at any term in a polynomial by any publicly accessible function. This allows wider scope for trying different implementations of polynomials where the terms may be represented in some implicit manner. On the other hand, there are many cases where an algorithm needs to iterate over the terms in a polynomial; some of these algorithms are inside PolyRing (i.e. the abstract class offers a suitable interface), but many will have to be outside for reasons of modularity and maintainability. Hence the need to have iterators which run through the terms in a polynomial.
The implementations in SparsePolyRing.C are all very simple: they just conduct some sanity checks on the function arguments before passing them to the PolyRing member function which will actually do the work.
Too many of the iterator functions are inline. Make them out of line, then use profiler to decide which should be inline.
PushFront and PushBack do not verify that the ordering criteria are
satisfied.
Verify the true need for myContent, myRemoveBigContent, myMulByCoeff,
myDivByCoeff, myMul (by pp). If the coeff ring has zero divisors then
myMulByCoeff could change the structure of the poly!
Verify the need for these member functions: myIsZeroAddLCs, myMoveLMToFront, myMoveLMToBack, myDeleteLM, myDivLM, myCmpLPP, myAppendClear, myAddClear, myAddMulLM, myReductionStep, myReductionStepGCD, myDeriv.
Should there be a RingHom accepting IndetImage (in case of univariate polys)?