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 0x2a34540
-- [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 = 4187
-- ncalls = 10
-- nloop = 29
-- nsaved = 0
-- -- used 0.010998 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 0x215a600 |
i9 : I = ideal random(R^1, R^{3:-3});
-- registering gb 4 at 0x2a34380
-- [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 0x2a341c0
-- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)m
removing gb 1 at 0x2a34700
ooo{8}(2)oo
-- number of (nonminimal) gb elements = 11
-- number of monomials = 267
-- ncalls = 10
-- nloop = 20
-- nsaved = 0
-- -- used 0.031995 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 0x215a300 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S)
7 3 1 2 5 2 3 1 2 9 6 2 7 2 10
o12 = ideal (-a + -a b + -a*b + b + -a c + -a*b*c + -b c + -a d + --a*b*d
6 2 3 3 5 7 5 9
-----------------------------------------------------------------------
2 9 2 1 2 6 2 2 1 2 3 1 2
+ 2b d + -a*c + -b*c + -a*c*d + 2b*c*d + -a*d + -b*d + c + -c d +
5 2 7 5 2 8
-----------------------------------------------------------------------
8 2 1 3 3 3 2 8 2 3 2 3 2 1 2
-c*d + -d , 4a + -a b + -a*b + 2b + 2a c + -a*b*c + 4b c + -a d +
5 2 5 9 2 2
-----------------------------------------------------------------------
2 1 2 2 2 3 5 1 2 3 2 4 3
-a*b*d + -b d + 2a*c + 3b*c + -a*c*d + -b*c*d + -a*d + --b*d + -c
5 2 4 7 3 10 3
-----------------------------------------------------------------------
9 2 4 2 7 3 3 3 5 2 2 2 7 3 7 2 5
+ --c d + -c*d + --d , --a + -a b + -a*b + -b + --a c + -a*b*c +
10 9 10 10 4 5 4 10 4
-----------------------------------------------------------------------
2 2 10 5 2 6 2 7 2 3 5 8 2
b c + a d + --a*b*d + -b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d +
9 8 7 3 4 6 5
-----------------------------------------------------------------------
2 2 3 2 2 1 3
-b*d + 5c + c d + c*d + -d )
7 8
o12 : Ideal of S
|
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J;
-- registering gb 6 at 0x2a34e00
-- [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 0.13198 seconds
1 39
o15 : Matrix S <--- S
|
i16 : selectInSubring(1,gens gb J)
o16 = | 228176594864827313524978572769797555273475699962871837686290465031192
-----------------------------------------------------------------------
576000000c27+3827986569020983554795728675708872122934345059948798168649
-----------------------------------------------------------------------
57860009305702400000c26d-
-----------------------------------------------------------------------
20290506750514376781862876375907486140751274433052682758449776853013574
-----------------------------------------------------------------------
45120000c25d2-137143282513457417077009891887018972541560381849785238244
-----------------------------------------------------------------------
3971698098774794240000c24d3+
-----------------------------------------------------------------------
26686529476158481968719174812997358902826280040804422327693303661274689
-----------------------------------------------------------------------
97427200c23d4+383125859452247668665122579481420478584351414137113462031
-----------------------------------------------------------------------
1180971114146762536960c22d5-
-----------------------------------------------------------------------
20835397179681066666822553153076487233283741803814209745525055273293391
-----------------------------------------------------------------------
01598720c21d6-648052612447883387705416704894789917841071515717959083800
-----------------------------------------------------------------------
7311091105164496186880c20d7-
-----------------------------------------------------------------------
19331244078769992464783181669900234528173982017569647240856814146811600
-----------------------------------------------------------------------
13889920c19d8+340058465426690470456657815217837521289937393356180553392
-----------------------------------------------------------------------
0352758032119090099640c18d9+
-----------------------------------------------------------------------
23721359952148103864264087640954736224146946754722013586898521928618988
-----------------------------------------------------------------------
84205592c17d10-35450090387460976958149298690875223430426018196587869415
-----------------------------------------------------------------------
2002566194366805030168c16d11-
-----------------------------------------------------------------------
19348442715336319561168872310175247775332789113003842611776798032884507
-----------------------------------------------------------------------
84759252c15d12-19933552318229164158394524397301176164477928640100694842
-----------------------------------------------------------------------
16098769574638029113748c14d13-
-----------------------------------------------------------------------
29863202677621682206883852573415536477279741578625565252699933677810247
-----------------------------------------------------------------------
8112540c13d14+173145436555835896741651537610429893756448639232344367859
-----------------------------------------------------------------------
93710544385896368616c12d15-
-----------------------------------------------------------------------
48129374260801208399034043677746855488312253430154867985584164868651932
-----------------------------------------------------------------------
8541134c11d16-240247971502834485416792975791627755754329240504971664233
-----------------------------------------------------------------------
290242574823469623049c10d17-
-----------------------------------------------------------------------
35957267489663859936671355507216039348572841049986545931741016502396263
-----------------------------------------------------------------------
493408c9d18-18399842962900756867859333567817357343430213037790397149233
-----------------------------------------------------------------------
49976798967363992c8d19+
-----------------------------------------------------------------------
16427742083865141678330675853873697827355061619592878589868288854000084
-----------------------------------------------------------------------
124960c7d20+87445035572515215874962443947854458514052239130937361975253
-----------------------------------------------------------------------
76626585676948496c6d21-
-----------------------------------------------------------------------
26692613977013662537958058728129547541809363542469310728946440195451325
-----------------------------------------------------------------------
55072c5d22-369815703919755578787354152001487147613814566699342764135870
-----------------------------------------------------------------------
284146633975808c4d23+33826935519550733311670054098814314233416586278996
-----------------------------------------------------------------------
92624000142297814106425600c3d24+
-----------------------------------------------------------------------
93135253974962680637249856352808701993804178467483960126234532706600312
-----------------------------------------------------------------------
3456c2d25-3507555703263293775341523301194184519841122879210166269481754
-----------------------------------------------------------------------
69312953871360cd26-2013522665488634247559473472734802403216771381524597
-----------------------------------------------------------------------
96664149793790609623040d27 |
1 1
o16 : Matrix S <--- S
|