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A.2.2 Groebner basis conversion
The performance of Buchberger's algorithm is
sensitive to the choice of monomial order. A Groebner basis
computation with respect to a less favorable order such as
the lexicographic ordering may easily run out of time or memory even
in cases where a Groebner basis computation with respect to a more efficient
order such as the degree reverse lexicographic ordering is very well
feasible. Groebner basis conversion algorithms and the Hilbert-driven
Buchberger algorithm are based on this observation:
SINGULAR provides implementations for the FGLM conversion algorithm
(which applies to zero-dimensional ideals only, see stdfglm) and
variants of the Groebner walk conversion algorithm (which works for
arbitrary ideals, See frwalk, grwalk_lib).
An implementation of the Hilbert-driven Buchberger
algorithm is accessible via the stdhilb command (see also std).
For the ideal below, stdfglm is more than 100 times
and stdhilb about 10 times faster than std .
| ring r =32003,(a,b,c,d,e),lp;
ideal i=a+b+c+d, ab+bc+cd+ae+de, abc+bcd+abe+ade+cde,
abc+abce+abde+acde+bcde, abcde-1;
int t=timer;
option(prot);
ideal j1=stdfglm(i);
==> [63:2]1(4)s2(3)s3(2)s4s(3)s5s(4)s(5)s(6)6-ss(7)s(9)s(11)-7-ss(13)s(15)s(1\
7)--s--8-s(16)s(18)s(20)s(23)s(26)-s(23)-------9--s(16)s10(19)s(22)s(25)-\
---s(24)--s11---------s12(17)s(19)s(21)------s(17)s(19)s(21)s13(23)s--s--\
---s(20)----------14-s(12)--------15-s(6)--16-s(5)--17---
==> (S:21)---------------------
==> product criterion:109 chain criterion:322
==> .....+....-...-..-+-....-...-..---...-++---++---....-...-++---.++-----------...------....-...------+--------+---.++------++++-+++----------------+---
==> vdim= 45
==> .............................................++--------------------------------------------+--------------------------------------------+--------------------------------------------+--------------------------------------------
timer-t;
==> 0
size(j1); // size (no. of polys) in computed GB
==> 5
t=timer;
ideal j2=stdhilb(i);
==> [31:1]1(4)s2(3)s3(2)s4s(3)s5s(4)s(5)s(6)6-ss(7)s(9)s(11)-7-ss(13)s(15)s(1\
7)--s--8-s(16)s(18)s(20)s(23)s(26)-s(29)-------9-s(25)s(28)--s(29)---s---\
----10-s(24)-------s(19)---11-s(17)s(19)s(21)-----s(18)-s(19)s12(21)s(23)\
s(26)-s(27)------s(23)----------13-s(15)-----------14-s(6)--15-s(5)--16--\
-
==> product criterion:88 chain criterion:650
==> [31:1]1(4)s2(3)s3(2)s4s(3)s5(5)s(8)s(10)s-ss6(12)s(15)s(17)s(20)s(21)s(22\
)s(24)s(25)--shhh7(23)s(25)s(27)s(29)s(31)-s(32)s(38)s(40)s(41)s(44)--shh\
hhhhhhhhh8(33)s(36)s(40)s(43)s(46)-s(47)s(50)--------shhhhhhhhhhhhh9(32)s\
(34)s(37)s(39)s(42)s(45)------s(42)s(45)shhhhhhhhhhh10(38)s(40)s(43)s(47)\
s(49)s(52)s(55)--s(56)------s(53)s(56)s(58)--shhhhhhhhhhhhhh11(45)s(47)s(\
50)s(53)s(56)s(59)s(61)---s-s(63)s(66)----s(65)-------shhhhhhhhhhhhhhhhh1\
2(44)s(47)s(50)s(51)s(53)s(56)s(58)s(61)s(63)s(66)s(69)s(72)--s(73)----sh\
hhhhhhhhhhhhhhhhhhhhhhhhh13(45)s(48)s(51)s(54)s(56)s(58)s(61)s(64)shhhhhh\
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh14(29)s(31)s(34)s(37)s(40)s(43)shhhhhhhhh\
hhhhhhhhhhhhhhh15(22)s(25)s(28)s(31)s(34)shhhhhhhhhhhhhhhhhh16(18)s(21)s(\
24)s(27)shhhhhhhhhhhhhhh17(15)s(18)s(21)s(24)shhhhhhhhhhhh18(15)s(18)s(21\
)s(24)shhhhhhhhhhhh19(14)s(17)s(20)shhhhhhhhhhhh20(11)s(14)s(17)shhhhhhhh\
h21(11)s(14)s(17)shhhhhhhhh22(11)s(14)s(16)shhhhhhhhh23(10)s(13)shhhhhhhh\
h24(7)s(10)shhhhhh25(7)s(10)shhhhhh26(7)s(10)shhhhhh27(7)s(10)shhhhhh28(7\
)s(10)shhhhhh29(7)s(10)shhhhhh30(7)s(9)shhhhhh31(6)shhhhhh[1023:2]32(3)sh\
hh33shhh34shhh35shhh36shhh37shhh38shhh39shhh40shhh41shhh42shhh43shhh44shh\
h45shhh46shhh47shhh48shhh49shhh50shhh51shhh52shhh53shhh54shhhhhh
==> product criterion:491 chain criterion:11799
==> hilbert series criterion:382
==> map[63:1]{2833:2944}----------...
timer-t;
==> 1
size(j2); // size (no. of polys) in computed GB
==> 158
// usual Groebner basis computation for lex ordering
t=timer;
ideal j0 =std(i);
==> [63:1]1(4)s2(3)s3(2)s4s(3)s5(5)s(4)s6(6)s(7)s(9)s(8)sss7(10)s(11)s(10)s(1\
1)s(13)s8(12)s(13)s(15)s.s(14).s.9.s(16)s(17)s(19)........10.s(20).s(21)s\
s..11.s(23)s(25).ss(27)...s(28)s(26)...12.s(25)sss(23)sss.......s(22)...1\
3.s(23)ssssssss(21)s(22)sssss(21)ss..14.ss(22)s.s.sssss(21)s(22)sss.s...1\
5.ssss(21)s(22)ssssssssss(21)s(22)sss16.ssssssss(21)s(22)sssssssssss17ss(\
21)s(22)ssssssssss(21)sss(22)ss(21)ss18(22)s(21)s(22)s.s..............19.\
sssss(21)ss(22)ssssssssss(21)s(22)s20.ssssssssss(21)s........21.s(22)ssss\
sssssssssss(21)s(22)ssss22ssssssssssss(21)s(22)sssssss23sssssssssss(21)s(\
22)ssssssss24ssssssssssss(21)s(22)sssssss25ssssssssss(21)s(22)sssssssss26\
ssssssssss(21)s(20)ssssssss27.sssssssss..........s28.ssssss.............2\
9.sssssssssssssssssss30sssssssssssssssssss31.sssssssssssssssssss32.ssssss\
ssssssssssssss33ssssssssssssssssssss34ssssssssssssssssssss35sssssssssssss\
sssssss36ssssssssssssssssssss37ssssssssssssssssssss38ssssssssssssssssssss\
39ssssssssssssssssssss40ssssssssssssssssssss41ssss---------------42-s(4)-\
-43-s44s45s46s47s48s49s50s51s52s53s54s55s56s
==> product criterion:1395 chain criterion:904
option(noprot);
timer-t;
==> 0
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