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 2 at 0x6351720
-- [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 = 4175
-- ncalls = 10
-- nloop = 29
-- nsaved = 0
-- -- used 0.016 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 3 at 0x5921580 |
i9 : I = ideal random(R^1, R^{3:-3});
-- registering gb 3 at 0x6351260
-- [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
-- registering gb 4 at 0x6351130
-- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber of (nonminimal) gb elements = 11
-- number of monomials = 266
-- ncalls = 10
-- nloop = 23
-- nsaved = 1
-- -- used 0.016 seconds
3 6 9
o10 = 1 - 3T + 3T - T
o10 : ZZ[T]
|
i11 : S = QQ[a..d,MonomialOrder=>Eliminate 2] -- registering polynomial ring 4 at 0x5921500 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S)
4 3 2 6 2 1 3 1 2 10 4 2 8 2
o12 = ideal (-a + a b + -a*b + -b + -a c + --a*b*c + -b c + -a d + 5a*b*d
3 5 4 7 7 5 7
-----------------------------------------------------------------------
3 2 2 7 2 4 5 2 4 2 1 3 2
+ --b d + 4a*c + -b*c + -a*c*d + 10b*c*d + -a*d + -b*d + -c + c d
10 5 3 3 7 2
-----------------------------------------------------------------------
1 2 3 1 3 2 2 8 3 8 2 2 7 2
+ -c*d + 4d , -a + a b + 5a*b + -b + -a c + a*b*c + b c + -a d +
3 4 3 5 3
-----------------------------------------------------------------------
2 5 2 1 2 1 5 5 2 5 2 1 3
9a*b*d + b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d + -b*d + -c +
3 2 4 3 6 2 4
-----------------------------------------------------------------------
10 2 9 2 1 3 4 3 2 8 2 7 3 2 1 6 2
--c d + -c*d + -d , -a + a b + -a*b + -b + 7a c + -a*b*c + -b c +
3 8 5 3 9 4 2 7
-----------------------------------------------------------------------
2 1 10 2 2 4 2 10 2
7a d + --a*b*d + --b d + 3a*c + -b*c + 8a*c*d + 4b*c*d + --a*d +
10 9 5 3
-----------------------------------------------------------------------
5 2 1 3 2 1 2 4 3
-b*d + -c + c d + -c*d + -d )
4 3 5 3
o12 : Ideal of S
|
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J;
-- registering gb 5 at 0x676fd10
-- [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)
--removing gb 0 at 0x63515f0
m{27}(1,3)m{28}(0,2)number of (nonminimal) gb elements = 39
-- number of monomials = 1050
-- ncalls = 46
-- nloop = 57
-- nsaved = 1
-- -- used 0.25 seconds
1 39
o15 : Matrix S <--- S
|
i16 : selectInSubring(1,gens gb J)
o16 = | 130254793832230002516245978058502286386868724706749061979110406571360
-----------------------------------------------------------------------
0c27+221955863685652702709804340078870871418900027315888577780456238642
-----------------------------------------------------------------------
838400c26d+464056983451079491353432188172633809448579073379883714113003
-----------------------------------------------------------------------
4667624242240c25d2-3010950698416360187669568418459267185778675737073837
-----------------------------------------------------------------------
91861182899350313681600c24d3-
-----------------------------------------------------------------------
14944726596313340733920387484529111430803694879043988068635257268890450
-----------------------------------------------------------------------
757376c23d4-19739049745250024674342046679193268336984820829128389673688
-----------------------------------------------------------------------
5038513506133703936c22d5-
-----------------------------------------------------------------------
12408101245352017081685303380305725986618713591422639561315819889784691
-----------------------------------------------------------------------
70794832c21d6-432126738294400354316580693068500452621479571818401713337
-----------------------------------------------------------------------
2932306907415079463360c20d7-
-----------------------------------------------------------------------
94357346905607815350098634206484288802356430404774736477644323226185595
-----------------------------------------------------------------------
38713296c19d8-176470121751171849135378839220133708921382671044460944732
-----------------------------------------------------------------------
21490003452948527635104c18d9-
-----------------------------------------------------------------------
45637087198040162112843358815890981026727333989773677036933252228693551
-----------------------------------------------------------------------
049429536c17d10-1220907497685455802789781781791727156187360343014780350
-----------------------------------------------------------------------
02870900254358475241107472c16d11-
-----------------------------------------------------------------------
22949252943688234309014175744065747103423022354425461871553840673603527
-----------------------------------------------------------------------
6107916276c15d12-280335000819769182002265619854191650175454210499458868
-----------------------------------------------------------------------
692664946240904024231709704c14d13-
-----------------------------------------------------------------------
22054730507462496807447592695589085473541391457742533138637619018582667
-----------------------------------------------------------------------
1187788540c13d14-110015947376784018502061655293036312135311280344939031
-----------------------------------------------------------------------
736544862411741559302311488c12d15-
-----------------------------------------------------------------------
34226821244015847516735400755896231251488344674807231775464013134009124
-----------------------------------------------------------------------
826459736c11d16-3086141158811918513268889080424274509564497702129445831
-----------------------------------------------------------------------
998260160068379524049924c10d17+
-----------------------------------------------------------------------
14736973466030492146347637844283873089912776459584873566776872448484143
-----------------------------------------------------------------------
603496285c9d18+27173963275675058464999535341999536120338663489033070302
-----------------------------------------------------------------------
064931303294909124932110c8d19+
-----------------------------------------------------------------------
25726864006468296264074747789704581819218432793508743959565831040468127
-----------------------------------------------------------------------
603912490c7d20+13987359801319012718846797597907029200589871626130282014
-----------------------------------------------------------------------
788271124769008664621780c6d21+
-----------------------------------------------------------------------
41151405000133121202489917446681042583072198802887522445409238698011332
-----------------------------------------------------------------------
94091800c5d22+691912734648881493584119852186636183670704002723480945610
-----------------------------------------------------------------------
763512422212635537500c4d23+
-----------------------------------------------------------------------
28684014492255360381998630637928624508462063359307367643438922478014318
-----------------------------------------------------------------------
7634500c3d24+1045522459664289456073149120848043750524286912950222989372
-----------------------------------------------------------------------
51728256785613047600c2d25-
-----------------------------------------------------------------------
14200196054387888822505236683572303405703134185189104657735900798325686
-----------------------------------------------------------------------
807500cd26-403390594517933740931279704350491244913876536864486875222577
-----------------------------------------------------------------------
3787288437924000d27 |
1 1
o16 : Matrix S <--- S
|