An installed Hilbert function will be used by Gröbner basis computations when possible.
Sometimes you know or are very sure that you know the Hilbert function. For example, in the following example, the Hilbert function of 3 random polynomials should be the same as the Hilbert function for a complete intersection.
i1 : R = ZZ/101[a..g]; |
i2 : I = ideal random(R^1, R^{3:-3});
o2 : Ideal of R
|
i3 : hf = poincare ideal(a^3,b^3,c^3)
3 6 9
o3 = 1 - 3T + 3T - T
o3 : ZZ[T]
|
i4 : installHilbertFunction(I, hf) |
i5 : gbTrace=3 o5 = 3 |
i6 : time poincare I
-- used 0. seconds
3 6 9
o6 = 1 - 3T + 3T - T
o6 : ZZ[T]
|
i7 : time gens gb I;
-- registering gb 3 at 0x2ce6e00
-- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)
-- number of (nonminimal) gb elements = 11
-- number of monomials = 4186
-- ncalls = 10
-- nloop = 29
-- nsaved = 0
-- -- used 0.013998 seconds
1 11
o7 : Matrix R <--- R
|
Another important situation is to compute a Gröbner basis using a different monomial order. In the example below
i8 : R = QQ[a..d]; -- registering polynomial ring 5 at 0x24f6000 |
i9 : I = ideal random(R^1, R^{3:-3});
-- registering gb 4 at 0x2ce6c40
-- [gb]
-- number of (nonminimal) gb elements = 0
-- number of monomials = 0
-- ncalls = 0
-- nloop = 0
-- nsaved = 0
--
o9 : Ideal of R
|
i10 : time hf = poincare I
removing gb 1 at 0x2b82000
-- registering gb 5 at 0x2ce6a80
-- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oo
-- number of (nonminimal) gb elements = 11
-- number of monomials = 267
-- ncalls = 10
-- nloop = 20
-- nsaved = 0
-- -- used 0.043993 seconds
3 6 9
o10 = 1 - 3T + 3T - T
o10 : ZZ[T]
|
i11 : S = QQ[a..d,MonomialOrder=>Eliminate 2] -- registering polynomial ring 6 at 0x2078c00 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S)
2 3 6 2 2 1 3 2 8 2 3 2 5
o12 = ideal (-a + -a b + 6a*b + -b + 5a c + -a*b*c + 9b c + -a d + -a*b*d
3 7 5 5 8 2
-----------------------------------------------------------------------
2 2 3 2 7 2 10 9 2 5 2 2 3 6 2
+ -b d + -a*c + -b*c + 6a*c*d + --b*c*d + -a*d + -b*d + -c + -c d
3 8 2 9 8 7 7 7
-----------------------------------------------------------------------
2 2 3 3 3 2 2 3 4 2 2 1 2 2
+ c*d + -d , a + -a b + 4a*b + b + -a c + -a*b*c + -b c + a d +
5 5 3 5 4
-----------------------------------------------------------------------
3 8 2 2 1 2 3 1 2 5 2 4 3
-a*b*d + -b d + a*c + -b*c + 2a*c*d + -b*c*d + -a*d + -b*d + -c +
5 5 3 5 3 6 3
-----------------------------------------------------------------------
2 2 2 5 3 3 7 2 1 2 2 3 4 2 4 3 2
9c d + -c*d + -d , a + -a b + -a*b + -b + -a c + -a*b*c + -b c +
7 4 6 2 3 3 7 8
-----------------------------------------------------------------------
5 2 3 7 2 9 2 10 2 6 6 2 1 2
-a d + -a*b*d + -b d + -a*c + --b*c + -a*c*d + 2b*c*d + -a*d + -b*d
4 5 9 7 7 5 7 3
-----------------------------------------------------------------------
8 3 2 9 2 3
+ -c + 5c d + -c*d + 6d )
5 8
o12 : Ideal of S
|
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J;
-- registering gb 6 at 0x2ce68c0
-- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
-- mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
-- {17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
-- {25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)
-- number of (nonminimal) gb elements = 39
-- number of monomials = 1051
-- ncalls = 46
-- nloop = 54
-- nsaved = 0
-- -- used 0.177973 seconds
1 39
o15 : Matrix S <--- S
|
i16 : selectInSubring(1,gens gb J)
o16 = | 633568315390036398986147947797313395612497492320430104868533900784663
-----------------------------------------------------------------------
983979397120c27+1062805681494116451373361111734848492392879712369834162
-----------------------------------------------------------------------
4290512726450511369609953280c26d+
-----------------------------------------------------------------------
10696607108823744427086128464109812939505053709243353721731148063399535
-----------------------------------------------------------------------
9414520934400c25d2+6049375308988139848428359471471241832352532056029419
-----------------------------------------------------------------------
13336471052785443764046258570240c24d3+
-----------------------------------------------------------------------
23880156303085702728074648359099896270941261768835381520628159592397397
-----------------------------------------------------------------------
59693952392320c23d4+605569055060404207354362139341006330175040016495555
-----------------------------------------------------------------------
5361858337731608582069731907908800c22d5+
-----------------------------------------------------------------------
14354526354870827148393325227364180205910960300808655687595445374796168
-----------------------------------------------------------------------
034113199998464c21d6+24730806544915194163616444284983454167542137092793
-----------------------------------------------------------------------
789054725411824273334941463801296304c20d7+
-----------------------------------------------------------------------
34949794423366697430227416375894473787935497432858974614453750503337775
-----------------------------------------------------------------------
389151582424832c19d8-57215047382564030342881754927535755554672508370263
-----------------------------------------------------------------------
11858444231748960049255022689949968c18d9+
-----------------------------------------------------------------------
13375476895909138658978037526938628897297477700036498183349967107044818
-----------------------------------------------------------------------
0114115964268896c17d10-
-----------------------------------------------------------------------
57593829401377391210721763600093962302461455769049753688703985287421368
-----------------------------------------------------------------------
462777762451636c16d11+1009269368498177094645908388008173964300041955090
-----------------------------------------------------------------------
41649649883633675683470442307506986592c15d12-
-----------------------------------------------------------------------
44541072445845914930772428548629782576171664704060957737421096131296498
-----------------------------------------------------------------------
404443794595500c14d13+6086918872082387238677612234264216491401513612807
-----------------------------------------------------------------------
777561178408736048405842861653009348c13d14+
-----------------------------------------------------------------------
29644722754515288646272439775335300132088382927832467268119340150113819
-----------------------------------------------------------------------
885712565608944c12d15-4176847665976881617260134718372650837197790969325
-----------------------------------------------------------------------
289227651379563357241042367207949260c11d16-
-----------------------------------------------------------------------
10393354924582195756736042322973246331070509378065317843813896311880598
-----------------------------------------------------------------------
93170146561905c10d17-21766657424294491539272328822622221266889916853309
-----------------------------------------------------------------------
6215900864443615206413286113133964c9d18+
-----------------------------------------------------------------------
34393366058964462588028741057236320539573857437701665515138258699962680
-----------------------------------------------------------------------
037854504006c8d19+29319106798784901200620642495959341677757670755579035
-----------------------------------------------------------------------
3203901322962058334631457680128c7d20-
-----------------------------------------------------------------------
18582274597349115838522125572614238309108726074246352238826984438864794
-----------------------------------------------------------------------
3387738474717c6d21+1093847128410955444963124139693095220242008546539025
-----------------------------------------------------------------------
0298474116001862857073931445244c5d22+
-----------------------------------------------------------------------
17451241928728290731560431455023500800612471038666924443951910392594771
-----------------------------------------------------------------------
531972222000c4d23-20790761386649501451726051327609587275993004450972966
-----------------------------------------------------------------------
98092546766957594357682031552c3d24-
-----------------------------------------------------------------------
97197909218516206376031429730611706810931836791311232111325692087071351
-----------------------------------------------------------------------
7427169280c2d25+9268291879955357739901958100746404423412883778003971107
-----------------------------------------------------------------------
3703135687920844320686080cd26+
-----------------------------------------------------------------------
23813432515425783512379722783838291419007542734242193679560812206310479
-----------------------------------------------------------------------
358197760d27 |
1 1
o16 : Matrix S <--- S
|