Operations

The permitted operations on ideals are: let I and J be two ideals of the same ring

• I+J -- the sum of two ideals
• I += J -- equivalent to I = I+J
• intersection(I, J) -- intersection of two ideals
• colon(I, J) -- the quotient of two ideals
• RingOf(I) -- the ring in which the ideal I resides
• NumGens(I) -- length of list of generators of I
• gens(I) -- a C++ vector<> of RingElems which generate I
• TidyGens(I) -- see also GBasis (below); returns a C++ vector of RingElems which generate I (this generating set is in some way "reduced", and will never contain a zero element)
  It is also possible to store some information about an ideal: (NOTE: making an incorrect assertion using these functions may lead to a program crash, wrong result, or poorer run-time performance)

   I->UserAssertsIsPrime()       to specify that I is known to be prime
   I->UserAssertsIsNotPrime()    to specify that I is known not to be prime
   I->UserAssertsIsMaximal()     to specify that I is known to be maximal
   I->UserAssertsIsNotMaximal()  to specify that I is known not to be maximal
  

Additional operations for ideals in a SparsePolyRing

• GBasis(I) -- returns a Groebner basis for I using the term-ordering of the polynomial ring in which I resides; the basis is stored inside I, so will not be recomputed in the future); equivalent to TidyGens.
• ReducedGBasis(I) -- like GBasis but guarantees that the result is a reduced basis.
• GBasisTimeout(I, Tmax) -- like GBasis but will throw an InterruptedByTimeout object if computation takes longer than Tmax seconds.

• LT(I) -- returns a monomial ideal, being the leading term ideal (also known as initial ideal)
• LF(I) -- returns a homogeneous ideal, being the leading form ideal (forms of maximum degree)
• IdealOfGBasis(I) -- returns a new ideal generated by the GBasis of I
• IdealOfMinGens(I) -- returns a new ideal equal to I but generated by a minimal set of gens (only for homogeneous ideals)
• PrimaryDecomposition(I) -- returns vector<ideal> only for square free monomial ideals or zero-dim ideals(for now)
• QuotientBasis(I) -- basis of the quotient as a K-vector space
• QuotientBasisSorted(I) -- same as QuotientBasis(I) but elements are sorted into increasing order.
• AlexanderDual(I) -- only for square free monomial ideals (for now)

• homog(h, I) -- returns the homogenized ideal, with homogenization with the indeterminate h, a RingElem, indeterminate in RingOf(I)
• MinPowerInIdeal(f, I) -- returns the smallest integer n such that power(f,n) is in I; returns -1 if polynomial f is not in the radical.

Writing new types of ideal

Anyone who writes a new type of ring class will have to consider writing a new type of ideal class to go with that ring. The ideal class must be derived from the abstract class IdealBase (and to be instantiable must offer implementations of all pure virtual functions). Be especially careful to update the data members IamPrime and IamMaximal in the non-const member functions (add, intersection, and colon).

Some guidance may be obtained from looking at the FieldIdealImpl class which implements ideals in a field (there are only two: ideal(0) and ideal(1)). See the file FieldIdeal.