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In the vesion before 2003,
facilities for computing Groebner bases are
`dp_gr_main()`

, `dp_gr_mod_main()`

and `dp_gr_f_main()`

.
To call these functions,
it is necessary to set several parameters correctly and it is convenient
to use a set of interface functions provided in the library file
‘`gr`’.
The facilities will be ready to use after you load the package by
`load()`

. The package ‘`gr`’ is placed in the standard library
directory of **Asir**.

In the current vesion, new functions such as `nd_gr`

, `nd_f4`

are available
and these function can compute Groebner bases more efficiently than old functions
(`nd_gr`

, `nd_gr_trace`

, `nd_f4`

, `nd_f4_trace`

, `nd_weyl_gr`

, `nd_weyl_gr_trace`

).

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.

In the current version, the functions corresponding to these three interfaces are provided
by `nd_gr`

, `nd_gr_trace`

.
See `nd_gr`

, `nd_gr_trace`

, `nd_f4`

, `nd_f4_trace`

, `nd_weyl_gr`

, `nd_weyl_gr_trace`

.

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