QBGenerator

© 2006 John Abbott
GNU Free Documentation License, Version 1.2



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User documentation for QBGenerator

The name QBGenerator derives from its intended use as a "quotient basis generator", that is a way of generating a factor closed (vector space) basis of power products for the quotient of a polynomial ring by a zero-dimensional ideal. It is used in the implementation of the FGLM and the Buchberger-Moeller algorithms -- in fact these are really the same algorithm (for computing a Groebner basis of an intersection of one or more zero-dimensional ideals).

Let P denote a polynomial ring (with coefficients in a field k), and let I be a zero-dimensional ideal in P. Then mathematically the quotient P/I is a finite dimensional vector space over k. We seek a basis QB for P/I which is a factor closed set of power products; i.e. if the power product t is in QB then any factor of t is in QB too. Groebner basis theory guarantees that such bases exist; actually it was first proved by Macaulay (a person, not a computer algebra system).

The elements of QB are determined one at a time, obviously starting with the trivial power product, 1. Moreover, at every stage the set of elements in the partially formed QB is factor closed, and this implies that only certain PPs are candidates for being adjoined to the QB (we call these "corners"). When a new element is adjoined to the QB new elements may appear in the corner set, these newly adjoined elements form the "new corner set" (this is always a subset of the corner set, and may be empty).

During the determination of the QB, some power products will be discovered which cannot be in the QB (usually based on a linear independence criterion). Such PPs form the "avoid" set: the QBGenerator will exclude all multiples of all elements of the avoid set from subsequent consideration.

Maintainer documentation for QBGenerator

Oh come on! Be reasonable! It's hot and humid here, I'm going on holiday tomorrow, and I can't remember what the code does. Nothing terribly fancy, I'm sure.

Bugs, Shortcomings and other ideas

myCornerPPIntoQB is definitely NOT EXCEPTION SAFE :-( How can I fix this without incurring excessive run-time overhead??? Maybe using splice instead of push_back would do the trick?

Might want to move declaration of it1 in myCornerPPIntoAvoidSet to before the call to push_back to be extra sure about exception safety.

Class QBGenerator could offer a ctor which accepts a (good) estimate of the dimension of the quotient, i.e. final number of elements in the QB. It could use this value to reserve space for myQBList.

How the blazes can I describe comprehensibly what QBGenerator does??

Documentation is incomplete.