**Type:**- Function
**Defined in:**- gb_reduce_all.c.trace
- Performs interreduction during the Gröbner bases algorithm.
`reducibles`contains polynomials, whose head term is reducible by one of the polynomials in`reducers`, which contains candidates to be adjoined to the basis.`basis`and`pairs`contain the current basis and pair set respectively.An element from

`reducibles`is reduced by polynomials from`reducers`and the basis. If this reduction yields a non-zero polynomial*p*, then*p*is adjoined to`reducers`and all polynomials in`reducers`and in the basis, whose head term can be reduced by*p*, change to`reducibles`. This can lead to a cascade of reductions until`reducibles`is empty. When this state is reached, the ``reducers'' are adjoined to the basis and the pair set is updated.A list containing the basis and the updated pair set is returned.

**See also:**- [1].

Thu Sep 3 14:50:07 MDT 1998