SparsePolyRing is an abstract class representing rings of polynomials; in particular, rings of sparse multivariate polynomials 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.
Currently there are four functions to create a polynomial ring:
NewPolyRing(CoeffRing, NumIndets)
CoeffRing
and having NumIndets
indeterminates. The PP ordering is StdDegRevLex
.
CoCoALib chooses automatically some names for the indeterminates
(currently the names are x[0], x[1], ... , x[NumIndets-1]).
NewPolyRing(CoeffRing, IndetNames)
CoeffRing
and having indeterminates whose names are given in IndetNames
(which
is of type vector<symbol>
). The PP ordering is StdDegRevLex
.
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
.
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.
A polynomial is viewed abstractly as a formal sum of ordered terms; 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 ring). The ordering is determined by the ordering on the power products: distinct terms in a polynomial must 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 in the given ordering.
We list here the operations available for an object of type SparsePolyRing
;
you should also consult the documentation in PolyRing.txt for operations on
more general sorts of polynomial ring.
Let P be an object of type SparsePolyRing
. Let R be an object of type ring
.
NumIndets(P)
-- the number of indeterminates in P.
CoeffRing(P)
-- the ring of coefficients of P.
PPM(P)
-- the PPMonoid of P.
GradingDim(P)
-- the dimension of the grading on P (may be 0).
IsSparsePolyRing(R)
-- returns true if the ring
R is indeed a SparsePolyRing
.
AsSparsePolyRing(R)
-- returns a SparsePolyRing
refering to the ring underlying R.
indets(P)
-- a const std::vector<RingElems>
whose i-th
element is the i-th indeterminate in P.
indet(P,i)
-- the i-th indet of P as a RingElem
.
IndetPower(P,i,n)
-- the n-th power of the i-th indet of P as a RingElem
.
In addition to the standard ring
and PolyRing
operations, elements of a
SparsePolyRing
may used in other functions.
Let P denote a SparsePolyRing
.
Let f denote a non-const element of P.
Let f1, f2 denote const elements of P.
Let expv be a vector<long>
of size equal to the number of indeterminates.
owner(f1)
-- the owner of f as a ring; NB to get the owner as
a SparsePolyRing use AsSparsePolyRing(owner(f1)).
NumTerms(f1)
-- the number of terms in f1.
LPP(f1)
-- the leading PP of f1; it is an element of PPM(P).
wdeg(f1)
-- the weighted degree of the leading PP of f1
(see [KR] Sec.4.3); error if f1 is 0.
NB result is of type CoCoA::degree
(see degree
).
(contrast with StdDeg(f1) and deg(f1) defined in PolyRing
)
CmpWDeg(f1, f2)
-- compare the weighted degrees of the LPPs of f1 and f2;
result is <0 =0 >0 according as deg(f1) < = > deg(f2)
CmpWDegPartial(f1, f2, i)
-- compare the first i weighted
degrees of the LPPs of f1 and f2;
result is <0 =0 >0 according as deg(f1) < = > deg(f2)
IsHomog(f)
-- says whether f is homogeneous wrt weighted degree.
IsHomogPartial(f,i)
-- says whether f is homogeneous wrt the
first i components of the weighted degree
homog(f, h)
-- returns f homogenized with indet h (requires
GrDim=1 and wdeg(h)=1)
NR(f, v)
-- returns the (normal) remainder of the Division
Algorithm by v. If v is a GBasis this is
the Normal Form
monomial(P,c,pp)
-- returns c*pp as an element of P where
c is in CoeffRing(P) and pp is in PPM(P).
monomial(P,c,expv)
-- returns c*x[0]^exps[0]*x[1]^exps[1]*... where
c is in CoeffRing(P), and x[i] are the indets of P.
-
BeginIter(f1)
-- a SparsePolyIter pointing to the first term in f1
EndIter(f1)
-- a SparsePolyIter pointing to one-past-the-last term in f1
Changing the value of f1 invalidates all iterators over f1.
!!Use these two functions with great care (c may be 0):
PushFront(f, c, expv)
-- add to f the term c*t
where t is the PP with exponent vector expv,
and ASSUMING that t > LPP(f) or f==0
PushBack(f, c, expv)
-- add to f the term c*t
where t is the PP with exponent vector expv,
and ASSUMING that t < t' for all t' appearing in f.
the corresponding member functions myPushFront/myPushBack
will not
check the validity of these assumpions: they should have a
CoCoA_ASSERT
to check in DEBUG mode.
Let i1 and i2 be two SparsePolyIter
s running over the same polynomial.
coeff(i1)
-- read-only access to the coeff of the current term
PP(i1)
-- read-only access to the pp of the current term
++i1
-- advance i1 to next term, return new value of i1
i1++
-- advance i1 to next term, return copy of old value of i1
i1 == i2
-- true iff i1 and i2 point to the same term;
throws CoCoA::ErrorInfo
with code ERR::MixedPolyIters
if i1 and i2 are over different polys.
i1 != i2
-- same as !(i1 == i2)
IsEnded(i1)
-- true iff i1 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, myMoveLM, myDeleteLM, myDivLM, myCmpLPP, myAppendClear, myAddClear, myAddMul, myReductionStep, myReductionStepGCD, myDeriv.
Should there be a RingHom accepting IndetImage (in case of univariate polys)?