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 0x81cad380
-- [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 = 4179
-- ncalls = 10
-- nloop = 29
-- nsaved = 0
-- -- used 0.029373 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 0x81eb5800 |
i9 : I = ideal random(R^1, R^{3:-3});
-- registering gb 4 at 0x81cad1c0
-- [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 5 at 0x81cad000
-- [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 0.019087 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 0x81eb5500 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S)
3 9 2 2 3 1 2 1 2 2 9
o12 = ideal (2a + -a b + a*b + 2b + -a c + 2a*b*c + -b c + a d + -a*b*d +
2 2 3 2
-----------------------------------------------------------------------
1 2 3 2 3 2 10 8 1 2 2 3 5 2
-b d + -a*c + --b*c + --a*c*d + -b*c*d + -a*d + 5b*d + c + -c d +
6 5 10 7 9 3 4
-----------------------------------------------------------------------
2 2 3 3 5 3 3 2 2 2 3 3 9 2 2 4 2 2
-c*d + -d , -a + -a b + -a*b + -b + -a c + -a*b*c + -b c + a d +
3 8 3 8 3 2 7 3 9
-----------------------------------------------------------------------
4 2 6 2 2 1 3 2 9 2 3 3
a*b*d + -b d + -a*c + 4b*c + -a*c*d + 5b*c*d + -a*d + -b*d + -c +
3 5 8 5 4 8
-----------------------------------------------------------------------
5 2 4 2 1 3 3 2 2 1 2 5 3 3 2 1 2
-c d + -c*d + -d , a + -a b + -a*b + -b + -a c + -a*b*c + 2b c +
2 3 2 5 2 7 5 2
-----------------------------------------------------------------------
8 2 1 2 2 2 5 2 1 9 2 2 2
-a d + -a*b*d + 3b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d + 3b*d
7 2 3 3 5 2 9
-----------------------------------------------------------------------
3 3 2 3 2 3 3
+ --c + 4c d + --c*d + -d )
10 10 5
o12 : Ideal of S
|
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J;
-- registering gb 6 at 0x82136e00
-- [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
removing gb 1 at 0x81cad540
-- {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 0.403043 seconds
1 39
o15 : Matrix S <--- S
|
i16 : selectInSubring(1,gens gb J)
o16 = | 812595474785400386641648810679446562982872592698683219198187601788928
-----------------------------------------------------------------------
c27+1850208566730622173337397133816573775770534889275315724779187887926
-----------------------------------------------------------------------
47680c26d+1352773504207709249695116158243739613358844964131993391266423
-----------------------------------------------------------------------
987167213568c25d2-30040986177825437860619054099042047629458623027610253
-----------------------------------------------------------------------
4193882810406978883584c24d3-
-----------------------------------------------------------------------
85916062097575174761044582172210165951368009609129928834588701887837804
-----------------------------------------------------------------------
26240c23d4-731112879447992352691929053412231076420768065891615452053830
-----------------------------------------------------------------------
33073055724360960c22d5-
-----------------------------------------------------------------------
31239117534340074396308416016985983154500004968371693995320606497066756
-----------------------------------------------------------------------
0056960c21d6-8293687246791126633895903472658342937868420964225114887351
-----------------------------------------------------------------------
52673382107614798720c20d7-
-----------------------------------------------------------------------
14844638131826467968969209145655825891989671086747856077118963161616800
-----------------------------------------------------------------------
07039040c19d8-182730933470358661662302566636564459891195237129307940942
-----------------------------------------------------------------------
5835896182877902506240c18d9-
-----------------------------------------------------------------------
14686321728157603541343421694273388087371572706335370249946895237611313
-----------------------------------------------------------------------
53078816c17d10-59836725550991023131599990235532728706527273776604922769
-----------------------------------------------------------------------
1749438307444435019680c16d11+
-----------------------------------------------------------------------
11559660528253417538795021419181368078859895591812379490232948479264612
-----------------------------------------------------------------------
7269584c15d12+270735287553825184006893592535366146577361127185683480047
-----------------------------------------------------------------------
362661451045039333328c14d13+
-----------------------------------------------------------------------
90432674108967672390646142711793574430715312180081049179727516155267801
-----------------------------------------------------------------------
582480c13d14-3952977849490063484675557754300286487015689786429858119421
-----------------------------------------------------------------------
2096212248292556640c12d15-
-----------------------------------------------------------------------
33605784804900512085043648157220663207188097842929438754607082796802824
-----------------------------------------------------------------------
363720c11d16-1933225184727368890152440399882453003943169660026821350203
-----------------------------------------------------------------------
017273052214110900c10d17+
-----------------------------------------------------------------------
31610593512882633653655198149375148401427401052552780959905110037353324
-----------------------------------------------------------------------
97700c9d18+212272826541403520472098312737295242921102519732280611015916
-----------------------------------------------------------------------
252875946369800c8d19+66274911607512832502772529158760373953087424518606
-----------------------------------------------------------------------
206411966108066930570950c7d20-
-----------------------------------------------------------------------
44495320600346242559330935209385167688218535485197819398121574936257669
-----------------------------------------------------------------------
025c6d21-34188396704735335545276247669229255677366888403116321375202807
-----------------------------------------------------------------------
098590394000c5d22-61606771515620784135333573856677424753595777610255730
-----------------------------------------------------------------------
70612823828117994000c4d23-
-----------------------------------------------------------------------
21294909068765014396146791083582106901659934128061725454884818124327025
-----------------------------------------------------------------------
00c3d24-510580644954777505645627596736226018802984218243730632608910112
-----------------------------------------------------------------------
023665125c2d25-46981271767042581459112944490411080945502532163831563265
-----------------------------------------------------------------------
588221449354500cd26-124329657065495161524943924352393071269609575036581
-----------------------------------------------------------------------
62177837622283005500d27 |
1 1
o16 : Matrix S <--- S
|