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## 8.2 Fundamental functions

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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