SparsePolyRing

© 2005,2007,2018 John Abbott, Anna M. Bigatti
GNU Free Documentation License, Version 1.2



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Examples

User documentation for SparsePolyRing

SparsePolyRing is an abstract class (inheriting from PolyRing) representing rings of polynomials; in particular, rings of sparse multivariate polynomials (e.g. written with *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; 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 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 (PPMonoidElem) in the given ordering.

See RingElem SparsePolyRing for operations on its elements.

Pseudo-constructors

Currently there are three functions to create a polynomial ring:

NewPolyRing(CoeffRing, IndetNames)
-- This creates a sparse polynomial ring with coefficients in 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)
-- This creates a sparse polynomial ring with coefficients in 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)
-- This creates a sparse polynomial ring with coefficients in 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)
-- sort of downcast the ring 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.

Query and cast

Let R be an object of type ring.

Operations on a SparsePolyRing

In addition to the standard PolyRing operations, a SparsePolyRing may be used in other functions.

Let P be an object of type SparsePolyRing.

Operations with SparsePolyIters

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.

Changing the value of f invalidates all iterators over f.

Examples

Maintainer documentation for SparsePolyRing

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.

Bugs, Shortcomings and other ideas

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)?