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 6.006e-6 seconds
3 6 9
o6 = 1 - 3T + 3T - T
o6 : ZZ[T]
|
i7 : time gens gb I;
-- registering gb 2 at 0xb407d10
-- [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 = 4171
-- ncalls = 10
-- nloop = 28
-- nsaved = 0
-- -- used 0.0175907 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 0xa9e6580 |
i9 : I = ideal random(R^1, R^{3:-3});
-- registering gb 3 at 0xb407850
-- [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 0xb407720
-- [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 = 267
-- ncalls = 10
-- nloop = 20
-- nsaved = 0
-- -- used 0.00753017 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 0xa9e6500 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S)
3 3 10 2 1 2 4 3 5 2 2 9 2 7 2
o12 = ideal (-a + --a b + -a*b + -b + -a c + -a*b*c + --b c + -a d +
5 3 5 3 9 3 10 3
-----------------------------------------------------------------------
1 1 2 10 2 7 2 4 9 9 2 8 2 3 3
-a*b*d + -b d + --a*c + -b*c + -a*c*d + -b*c*d + --a*d + -b*d + -c
2 4 3 5 5 7 10 3 8
-----------------------------------------------------------------------
5 2 1 2 3 3 3 1 2 7 2 3 3 1 2 9 2
+ -c d + -c*d + d , -a + -a b + -a*b + -b + -a c + 8a*b*c + -b c +
6 6 4 3 3 2 2 4
-----------------------------------------------------------------------
10 2 4 2 1 2 7 2 6 2 8 2
--a d + a*b*d + -b d + -a*c + --b*c + -a*c*d + b*c*d + 7a*d + -b*d
3 7 2 10 7 9
-----------------------------------------------------------------------
3 1 2 2 3 3 1 3 1 2 4 2 7 3 1 2
+ 8c + -c d + 7c*d + -d , -a + --a b + -a*b + -b + -a c + 9a*b*c +
3 4 2 10 5 3 6
-----------------------------------------------------------------------
2 5 2 1 5 2 10 2 2 2 1 1 2
2b c + -a d + -a*b*d + -b d + --a*c + -b*c + a*c*d + -b*c*d + -a*d +
7 2 4 9 3 8 3
-----------------------------------------------------------------------
5 2 3 2 4 2 8 3
-b*d + 5c + 2c d + -c*d + -d )
4 7 7
o12 : Ideal of S
|
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J;
-- registering gb 5 at 0xb407390
-- [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
--removing gb 0 at 0xb407be0
{28}(0,2)number of (nonminimal) gb elements = 39
-- number of monomials = 1051
-- ncalls = 46
-- nloop = 54
-- nsaved = 0
-- -- used 0.227361 seconds
1 39
o15 : Matrix S <--- S
|
i16 : selectInSubring(1,gens gb J)
o16 = | 521357549177464751589063726149291012606636689620769586961370394717713
-----------------------------------------------------------------------
0461687552000c27+771630390425268006513194313897440754530699965568800747
-----------------------------------------------------------------------
39781585024623575375880780800c26d+
-----------------------------------------------------------------------
48189724745659353897217521458488771173549335923481313538143260290072761
-----------------------------------------------------------------------
2741150004480c25d2+1259692219600095182409875805010955697083264838575732
-----------------------------------------------------------------------
546969923988096813672599940470272c24d3+
-----------------------------------------------------------------------
30127609590592071534969774636110010847105641293401508364391329533608327
-----------------------------------------------------------------------
97519059928832c23d4+407813655195767354733908592568781106181674588286032
-----------------------------------------------------------------------
3139737181379387766107147505424896c22d5+
-----------------------------------------------------------------------
43981687054259419509247861671426434546243034684133731276195763782331458
-----------------------------------------------------------------------
32431075953216c21d6+855716689452710909178569680727022518551840940490573
-----------------------------------------------------------------------
914598346285759122665492266308288c20d7-
-----------------------------------------------------------------------
52408641364299766777643979295898857398489962005791548398431122414220416
-----------------------------------------------------------------------
35048425370080c19d8-102395158326752409184630487489746153020165012147110
-----------------------------------------------------------------------
88020064618775122911082474504366528c18d9-
-----------------------------------------------------------------------
98004103825536325523769092330314514984545164463645707410105257500680093
-----------------------------------------------------------------------
31771655775364c17d10-31447503434115477519504108357012221404679223167862
-----------------------------------------------------------------------
1066670516399010046822181883845334c16d11+
-----------------------------------------------------------------------
53932483855337784618825154900134485594929982501524087637285918158367075
-----------------------------------------------------------------------
72122535365923c15d12+14851287832155100801655170341035611605602671912769
-----------------------------------------------------------------------
699971852514422014903013889734116008c14d13-
-----------------------------------------------------------------------
47255130748592915642996610262697431428611468431020341682802453104440007
-----------------------------------------------------------------------
8301762311840c13d14-722546219626382677031951474049730714941870429188514
-----------------------------------------------------------------------
5138490315796225774569289695340320c12d15-
-----------------------------------------------------------------------
46432016004228975221360657121418910526514740676723071344537223273872325
-----------------------------------------------------------------------
52997083407600c11d16-57077419337341163773958821169275999253959126788798
-----------------------------------------------------------------------
91761925182933337985586701909092800c10d17+
-----------------------------------------------------------------------
89226122812931244034165778623231173272563531740841738708663600641124802
-----------------------------------------------------------------------
36857963020000c9d18+403352214847878394899869752988862181424213927219487
-----------------------------------------------------------------------
638797879539698801042871780192000c8d19+
-----------------------------------------------------------------------
67779623992759414597430535255822289609143317459656829184354484052657210
-----------------------------------------------------------------------
54878301904000c7d20-160032794936432888870893805834307444759330165211261
-----------------------------------------------------------------------
6714242680508239505475315069320000c6d21+
-----------------------------------------------------------------------
15222059550308378327685971676192058771786284261687330359918279003013209
-----------------------------------------------------------------------
03255345280000c5d22-112751497883560608943553596385305612556108319705901
-----------------------------------------------------------------------
0763375500205103736096163940000000c4d23+
-----------------------------------------------------------------------
12339674479353824676531592350918934759025033324484665639667953031159084
-----------------------------------------------------------------------
1417804800000c3d24-2470448103523205343893993060089723919958148566193034
-----------------------------------------------------------------------
50618826130489491503788144000000c2d25+
-----------------------------------------------------------------------
37616606766712756544606455872793791144605707162327916420265550760310510
-----------------------------------------------------------------------
95040000000cd26-1928760909663829306863251326541379688329238929822180648
-----------------------------------------------------------------------
0362819200125544363200000000d27 |
1 1
o16 : Matrix S <--- S
|