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There are many functions and options defined in the package ‘`gr`’.
Usually not so many of them are used. Top level functions for Groebner
basis computation are the following three functions.

In the following description, `plist`, `vlist`, `order`
and `p` stand for a list of polynomials, a list of variables
(indeterminates), a type of term ordering and a prime less than
`2^27`

respectively.

`gr(`

`plist`,`vlist`,`order`)-
Function that computes Groebner bases over the rationals. The algorithm is Buchberger algorithm with useless pair elimination criteria by Gebauer-Moeller, sugar strategy and trace-lifting by Traverso. For ordinary computation, this function is used.

`hgr(`

`plist`,`vlist`,`order`)-
After homogenizing the input polynomials a candidate of the \gr basis is computed by trace-lifting. Then the candidate is dehomogenized and checked whether it is indeed a Groebner basis of the input. Sugar strategy often causes intermediate coefficient swells. It is empirically known that the combination of homogenization and supresses the swells for such cases.

`gr_mod(`

`plist`,`vlist`,`order`,`p`)-
Function that computes Groebner bases over GF(

`p`). The same algorithm as`gr()`

is used.

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