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A.2.1 groebner and std
The basic version of Buchberger's algorithm leaves a lot
of freedom in carrying out the computational process. Considerable
improvements are obtained by implementing criteria
for reducing the number of S-polynomials to be actually considered
(e.g., by applying the product criterion or the chain criterion).
We refer to Cox, Little, and O'Shea [1997], Chapter 2 for more details
and references on these criteria and on further strategies for improving the
performance of Buchberger's algorithm (see also Greuel, Pfister [2002]).
SINGULAR's implementation of Buchberger's algorithm is available via
the std command ('std' referring to standard basis). The
computation of reduced Groebner and standard bases may be forced by setting
option(redSB) (see option).
However, depending on the monomial ordering of the active basering, it may be
advisable to use the groebner command instead. This command is
provided by the SINGULAR library standard.lib which
is automatically loaded when starting a SINGULAR session. Depending
on some heuristics, groebner either refers to the std
command (e.g., for rings with ordering dp), or to one of the
algorithms described in the sections Groebner basis conversion,
slim Groebner bases.
For information on the heuristics behind groebner, see the
library file standard.lib (see also Procedures and libraries).
We apply the commands std and groebner to compute a
lexicographic Groebner basis for the ideal of cyclic roots over the
basering with 6 variables (see Cyclic roots). We set
option(prot) to make SINGULAR display some information on
the performed computations (see option for an interpretation of
the displayed symbols). For long running computations, it is always
recommended to set this option.
| | LIB "poly.lib";
ring r=32003,(a,b,c,d,e,f),lp;
ideal I=cyclic(6);
option(prot);
int t=timer;
system("--ticks-per-sec", 100); // give time in 1/100 sec
ideal sI=std(I);
==> [1023:1]1(5)s2(4)s3(3)s4s(4)s5(6)s(9)s(11)s(14)s(17)-s6s(19)s(21)s(24)s(2\
7)s(30)s(33)s(35)s(38)s(41)ss(42)-s----s7(41)s(43)s(46)s(48)s(51)s(54)s(5\
6)s(59)s(62)s(63)s(65)s(66)s(68)s(70)s(73)s(75)s(78)---ss(81)-----------s\
(73)--------8-s(66)s(69)s(72)s(75)s(77)s(80)s(81)s(83)s(85)s(88)s(91)s(93\
)s(96)s(99)s(102)s(105)s(107)s(110)s(113)-------------(100)-----------s(1\
01)s(108)s(110)----------(100)--------9-s(94)s(97)s(99)s(84)s(74)s(77)s(8\
0)---ss(83)s(86)s(73)s(76)s10(78)s(81)s(82)s(84)s(86)s(89)s(92)s(94)s(97)\
s(100)s(103)s(82)s(84)s(86)s(89)s(92)s(95)s11(98)s(87)s(90)s(93)s(95)s(98\
)s(101)s(104)----(100)---12-s(99)s(90)s(93)s(92)---------s(86)-----------\
13-s(74)s(77)s(79)s(82)s(85)s(88)------------------14-s(64)s(67)ss(70)s(7\
3)s(77)s(81)-----------------------15-s(57)s(65)s(68)ss(71)--------------\
---s(57)----16-s(55)ss(56)------------------------17-s(34)s(32)--------18\
-s(26)s(28)s-----19-s(25)s(28)s(31)------20-s(27)s(30)s(35)-------21-s(23\
)s(26)------22-s(22)------23-s(15)24-s(17)-s(19)--25-s(18)s(19)s26-s(21)-\
--------27-s(11)28-s(13)--29-s(12)-30--s--31-s(11)---32-s33(7)s(10)---34-\
s-35----36-s37(6)s38s39s40---42-s43(5)s44s45--48-s49s50s51---54-s55(4)--6\
7-86-
==> product criterion:664 chain criterion:2844
timer-t; // used time (in 1/100 secs)
==> 11
size(sI);
==> 17
t=timer;
sI=groebner(I);
==> compute hilbert series with std in ring (ZZ/32003),(a,b,c,d,e,f,@),(dp(7)\
,C)
==> weights used for hilbert series: 1,1,1,1,1,1,1
==> [511:1]1(5)s2(4)s3(3)s4ss5(4)s(5)s(7)-s6(8)s(9)s(11)s(13)s(16)s(18)s(21)-\
-s7(22)s(23)s(24)s(27)s(29)s(31)s(32)s(35)-s(37)s(40)s(42)s(44)s(45)--s(4\
6)s(48)-----8-s(44)s(47)s(50)s(52)s(55)s(57)s(59)s(61)-s(63)----s(62)----\
s(61)s(64)-s(66)-----------s(58)-------9-s(53)s(56)s(59)s(62)s(65)s(68)s(\
71)s(74)s(77)s(80)s(83)s(86)s(90)s(95)s(102)s(108)--------(100)----------\
------------s(81)---10-s(83)s(88)s(90)s(94)s(99)s(104)s(109)s(114)-s(116)\
s(121)s(126)s(128)s(132)--------------------------------(100)------------\
--11-s(87)---------------------------------------12-s(50)--------13-s(44)\
s(47)s(51)s(55)-------------14-s(45)s(48)s(51)s(55)s(58)s(61)s(64)s(67)s(\
70)--------------------15-s(52)s(55)s(58)s(61)s(64)s(67)s(70)s(73)s(76)s(\
79)s(82)-------------------------------------16--------------------------\
------------------17-
==> product criterion:284 chain criterion:4184
==> std with hilb in (ZZ/32003),(a,b,c,d,e,f,@),(lp(6),dp(1),C)
==> [511:1]1(98)s2(97)s3(96)s4s(97)-s5(98)s(101)s(103)s(106)s(109)---s6(107)s\
(109)s(111)s(114)s(117)s(120)s(123)s(125)s(128)s(131)ss(132)-s----------s\
7(125)s(127)s(130)s(132)s(135)s(138)s(140)s(143)s(146)s(147)s(149)s(150)s\
(152)s(154)s(157)s(159)s(162)---ss(165)-----------shhhhhhhhhhhhhhhhhhhhhh\
h8(134)s(136)s(139)s(142)s(145)s(147)s(150)s(151)s(153)s(155)s(158)s(161)\
s(163)s(166)s(169)s(172)s(175)s(177)s(180)s(183)------------------------s\
(171)s(178)shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh9(147)s(150)s(153)s(155)s(18\
1)s(184)s(187)s(190)s(203)s(208)s(213)s(217)s(218)s(220)s(222)s(225)---s-\
s(226)-----------s(219)---------shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh\
hhhhhhhhh10(163)s(166)s(168)s(171)s(177)s(180)s(183)s(186)shhhhhhhhhhhhhh\
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh11(125)s(128)s(130)\
s(133)s(136)shhhhhhhhhhhhhhhhhhhhhhhhhhhhh12(110)s(113)s(120)s(123)s(127)\
-------------shhhhhhhhhhhhhhh13(102)s(106)s(109)s(111)s(114)s(117)----shh\
hhhhhhhhhhhhhhhhhhhhhhhhhhhhh14(85)s(90)s(93)s(97)s(100)s(103)---(100)-s(\
103)shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh15(68)s(72)s(75)s(79)s(85)---\
-shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh16(42)s(45)s(49)shhhhhhhhhhh\
hhhhhhhhhhh17(34)s(37)shhhhhhhhhhhhhh18(27)s(30)s(32)-shhhhhhhhh19(26)s(2\
9)s(32)shhhhhhhhhhhhh20(22)s(25)s(28)shhhhhhhhhhhhh21(20)s(26)shhhhhhhhhh\
hh22(18)shhhhhhhhh23(12)shhhhh24(11)s(14)-shhhh25(13)s(18)-s(21)shhhhhhh2\
6(18)shhhhhhhhhhhh27(9)shhhhh28(8)shhhh29(7)shhhh30(8)-shhh31shhhhh32(7)s\
hhhh33shhhh34(6)shhhhhhhh36(2)s37(6)shhhh38shhhh39shhhhhhhh42(2)s43(5)shh\
hh44shhhhhhhh48s49shhhh50shhhhhhhh54shhhhh
==> product criterion:720 chain criterion:11620
==> hilbert series criterion:532
==> dehomogenization
==> simplification
==> imap to ring (ZZ/32003),(a,b,c,d,e,f),(lp(6),C)
timer-t; // used time (in 1/100 secs)
==> 4
size(sI);
==> 17
option(noprot);
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