An installed Hilbert function will be used by Groebner 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 0x11d1b700 -- [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 = 4193 -- -- used 0.04512 seconds 1 11 o7 : Matrix R <--- R |
Another important situation is to compute a Groebner basis using a different monomial order. In the example below
i8 : R = QQ[a..d]; |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 4 at 0x11d1b540 -- [gb]number of (nonminimal) gb elements = 0 -- number of monomials = 0 -- o9 : Ideal of R |
i10 : time hf = poincare I -- registering gb 5 at 0x11d1b380 -- [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 -- -- used 0.018434 seconds 3 6 9 o10 = 1 - 3T + 3T - T o10 : ZZ [T] |
i11 : S = QQ[a..d,MonomialOrder=>Eliminate 2] o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 5 3 9 2 7 2 5 3 2 2 6 2 7 2 o12 = ideal (-*a + -*a b + -*a*b + -*b + -*a c + -*a*b*c + 8b c + -*a d + 4 2 6 6 5 5 6 ----------------------------------------------------------------------- 3 6 2 1 2 9 2 10 3 7 2 -*a*b*d + -*b d + -*a*c + -*b*c + --*a*c*d + -*b*c*d + -*a*d + 4 7 4 8 3 4 4 ----------------------------------------------------------------------- 8 2 1 3 4 2 9 2 1 3 3 2 1 2 3 3 -*b*d + -*c + -*c d + --*c*d + -*d , a + 8a b + -*a*b + -*b + 7 2 3 10 4 9 2 ----------------------------------------------------------------------- 4 2 2 2 2 1 4 2 8 2 1 2 -*a c + 4a*b*c + -*b c + 4a d + -*a*b*d + -*b d + -*a*c + -*b*c + 7 3 2 9 9 9 ----------------------------------------------------------------------- 2 3 2 2 1 3 3 2 1 2 3 3 4 3 -*a*c*d + 8b*c*d + -*a*d + b*d + -*c + -*c d + -*c*d + -*d , -*a + 3 4 7 5 9 7 3 ----------------------------------------------------------------------- 9 2 3 2 1 3 2 2 5 7 2 9 2 7 -*a b + -*a*b + -*b + -*a c + -*a*b*c + -*b c + -*a d + -*a*b*d + 5 7 9 3 9 4 8 2 ----------------------------------------------------------------------- 4 2 1 2 9 2 5 9 2 2 2 1 3 -*b d + -*a*c + -*b*c + -*a*c*d + -*b*c*d + 2a*d + -*b*d + --*c + 5 3 4 7 7 3 10 ----------------------------------------------------------------------- 2 7 2 7 3 c d + -*c*d + -*d ) 6 4 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 6 at 0x11d1b1c0 -- [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 removing gb 1 at 0x11d1ba80 {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 -- -- used 0.396755 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) -- [gb]number of (nonminimal) gb elements = 39 -- number of monomials = 1051 -- o16 = | 109951076075589603607379639092920107191077842047433285281607021558810 ----------------------------------------------------------------------- 049883707801600000c27-4686924695125355562559234266999248755751350152530 ----------------------------------------------------------------------- 37290637037940007124467934215536640000c26d- ----------------------------------------------------------------------- 12888291820625910834073001746119126860044705851357375766895168428954565 ----------------------------------------------------------------------- 25580988547072000c25d2+ ----------------------------------------------------------------------- 95311851768318039167855679599244940831952767216781272775431316971641562 ----------------------------------------------------------------------- 02287755783372800c24d3- ----------------------------------------------------------------------- 19720663914619462529571958494452271162838828009511110238198796727099147 ----------------------------------------------------------------------- 391747155520716800c23d4+ ----------------------------------------------------------------------- 22631931494755499156618325247568305540815947458458538169339088046843149 ----------------------------------------------------------------------- 170261940368793600c22d5+ ----------------------------------------------------------------------- 79180770775424934311786219321813577090903288687488746822676303042658723 ----------------------------------------------------------------------- 947151877874736640c21d6- ----------------------------------------------------------------------- 70852865029503702076931394367454782497013534306955737456314674042082334 ----------------------------------------------------------------------- 5481124281311806720c20d7- ----------------------------------------------------------------------- 63056696006154486460476392098586257274854299612350762644665814420335589 ----------------------------------------------------------------------- 5854949894261658304c19d8- ----------------------------------------------------------------------- 17663164893499106882017960061881991268977192484964368197064177275083427 ----------------------------------------------------------------------- 60407072362578802280c18d9- ----------------------------------------------------------------------- 23706232902010981845080453180190524362336194827712777118643619757153629 ----------------------------------------------------------------------- 94585345774342091240c17d10- ----------------------------------------------------------------------- 36248816708024852747912681911690706186309968240027100522620311660437244 ----------------------------------------------------------------------- 46325461005320349600c16d11- ----------------------------------------------------------------------- 38863052051014026424587160206641671418633853757001941879862235459260343 ----------------------------------------------------------------------- 49271813881604730150c15d12- ----------------------------------------------------------------------- 42959781501395150748401784193043644329064842704838213752292269175783430 ----------------------------------------------------------------------- 4388677844758267500c14d13+ ----------------------------------------------------------------------- 75742107456609238709616906536179732666715184557906163317363104456553863 ----------------------------------------------------------------------- 59558924864900861250c13d14+ ----------------------------------------------------------------------- 56893630198252618257292705320362082297010112147121791406470851826037924 ----------------------------------------------------------------------- 39139722553157791875c12d15- ----------------------------------------------------------------------- 40438448888370763841981288825165492438896938677878008146699569009349293 ----------------------------------------------------------------------- 36101128769251225000c11d16- ----------------------------------------------------------------------- 75486844549582972533333155477247890096429994635256520997565687641033889 ----------------------------------------------------------------------- 60351992505345662500c10d17- ----------------------------------------------------------------------- 35055422859066880578033780619139560310564026128104277379786246946290189 ----------------------------------------------------------------------- 17189504723840537500c9d18- ----------------------------------------------------------------------- 13105445664092883265927311090374317649773893375042661029784648290644147 ----------------------------------------------------------------------- 77285136774884375000c8d19+ ----------------------------------------------------------------------- 19735417636311826225733525052852991499590850140339844318804800179871016 ----------------------------------------------------------------------- 9840400773925000000c7d20+ ----------------------------------------------------------------------- 20427445564098785302443614268027449606508541419343696274910467558869112 ----------------------------------------------------------------------- 755446428012500000c6d21- ----------------------------------------------------------------------- 25532316737975998333943559832484179655919920706895946123659828966358680 ----------------------------------------------------------------------- 760982658312500000c5d22- ----------------------------------------------------------------------- 54476682784538556757997883635539857254735345641479894160372762788175869 ----------------------------------------------------------------------- 85121553750000000c4d23- ----------------------------------------------------------------------- 31813421210247519349180671637350302853585189848821503364452698068174585 ----------------------------------------------------------------------- 7383125000000000c3d24-5394190915687107576430451947341829022667216566107 ----------------------------------------------------------------------- 343532021994385600075028750000000000c2d25+ ----------------------------------------------------------------------- 11971077024379141772176603915187908057028816122601812328775518067594270 ----------------------------------------------------------------------- 000000000000cd26-253515663350606316265600092747204231107338737527376715 ----------------------------------------------------------------------- 94905259844757031250000000d27 | 1 1 o16 : Matrix S <--- S |