nd_gr, nd_gr_trace, nd_f4, nd_f4_trace, nd_weyl_gr, nd_weyl_gr_trace

nd_gr(plist,vlist,p,order)

nd_gr_trace(plist,vlist,homo,p,order)

nd_f4(plist,vlist,modular,order)

nd_f4_trace(plist,vlist,homo,p,order)

nd_weyl_gr(plist,vlist,p,order)

nd_weyl_gr_trace(plist,vlist,homo,p,order)

:: Groebner basis computation (built-in functions)

return

list

plist vlist

list

order

number, list or matrix

homo

flag

modular

flag or prime

• These functions are new implementations for computing Groebner bases.
• nd_gr executes Buchberger algorithm over the rationals if p is 0, and that over GF(p) if p is a prime.
• nd_gr_trace executes the trace algorithm over the rationals. If p is 0 or 1, the trace algorithm is executed until it succeeds by using automatically chosen primes. If p a positive prime, the trace is comuted over GF(p). If the trace algorithm fails 0 is returned. If p is negative, the Groebner basis check and ideal-membership check are omitted. In this case, an automatically chosen prime if p is 1, otherwise the specified prime is used to compute a Groebner basis candidate. Execution of nd_f4_trace is done as follows: For each total degree, an F4-reduction of S-polynomials over a finite field is done, and S-polynomials which give non-zero basis elements are gathered. Then F4-reduction over Q is done for the gathered S-polynomials. The obtained polynomial set is a Groebner basis candidate and the same check procedure as in the case of nd_gr_trace is done.
• nd_f4 executes F4 algorithm over Q if modular is equal to 0, or over a finite field GF(modular) if modular is a prime number of machine size (<2^29).
• nd_weyl_gr, nd_weyl_gr_trace are for Weyl algebra computation.
• Each function cannot handle rational function coefficient cases.
• In general these functions are more efficient than dp_gr_main, dp_gr_mod_main, especially over finite fields.

[38] load("cyclic")$
[49] C=cyclic(7)$
[50] V=vars(C)$
[51] cputime(1)$
[52] dp_gr_mod_main(C,V,0,31991,0)$
26.06sec + gc : 0.313sec(26.4sec)
[53] nd_gr(C,V,31991,0)$
ndv_alloc=1477188
5.737sec + gc : 0.1837sec(5.921sec)
[54] dp_f4_mod_main(C,V,31991,0)$  
3.51sec + gc : 0.7109sec(4.221sec)
[55] nd_f4(C,V,31991,0)$
1.906sec + gc : 0.126sec(2.032sec)

References

section dp_ord, section dp_gr_flags, dp_gr_print, section Controlling Groebner basis computations

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