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 0x17af390 -- [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 = 4168 -- ncalls = 10 -- nloop = 29 -- nsaved = 0 -- -- used 0.018984 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 0x17667e0 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 4 at 0x1913e40 -- [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 0x1913d10 -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)m removing gb 1 at 0x17af260 ooo{8}(2)oo -- number of (nonminimal) gb elements = 11 -- number of monomials = 267 -- ncalls = 10 -- nloop = 20 -- nsaved = 0 -- -- used 0.058585 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 0x1585f30 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 6 3 4 2 3 2 2 3 7 2 1 2 2 2 o12 = ideal (-a + -a b + -a*b + -b + -a c + 6a*b*c + --b c + -a d + 5 5 2 3 3 10 7 ----------------------------------------------------------------------- 10 6 2 4 2 1 2 5 5 1 2 1 2 5 3 --a*b*d + -b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d + -b*d + -c 3 5 5 4 6 9 3 6 3 ----------------------------------------------------------------------- 2 5 2 9 3 3 3 2 2 3 3 3 2 7 2 2 + 2c d + -c*d + --d , -a + 4a b + a*b + -b + -a c + -a*b*c + -b c + 3 10 2 2 2 6 9 ----------------------------------------------------------------------- 2 2 5 2 2 3 2 5 4 1 2 2 5a d + -a*b*d + -b d + 4a*c + -b*c + -a*c*d + -b*c*d + -a*d + 8b*d 3 9 4 3 3 5 ----------------------------------------------------------------------- 7 3 4 2 2 2 1 3 5 3 3 2 1 2 3 4 2 10 + --c + -c d + -c*d + -d , -a + -a b + -a*b + b + -a c + --a*b*c + 10 3 7 8 6 2 2 5 3 ----------------------------------------------------------------------- 6 2 2 10 2 1 2 9 2 4 5 8 2 -b c + 3a d + 5a*b*d + --b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d 7 7 2 8 3 7 9 ----------------------------------------------------------------------- 5 2 3 5 2 4 2 4 3 + -b*d + 2c + -c d + -c*d + -d ) 8 2 7 9 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 6 at 0x1913980 -- [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 0.732338 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 902789818494588265793188817874962007487070867710992449181592733333180 ----------------------------------------------------------------------- 7168c27+964663043641956021515913640341087673119254045819159778931449541 ----------------------------------------------------------------------- 57772753376c26d+1411936630320430960681511773423293537713086636889459724 ----------------------------------------------------------------------- 36682571904091445520c25d2- ----------------------------------------------------------------------- 93596700579746795399738147513156114634695450593550244806909136688158806 ----------------------------------------------------------------------- 5800c24d3-2183362713097226939926355557404010074752482309849925671407195 ----------------------------------------------------------------------- 321184099085160c23d4+68700885846997046711148323564078730259537323423113 ----------------------------------------------------------------------- 66125787990419518292845996c22d5+ ----------------------------------------------------------------------- 23132989042384126732295420242574410370858155161894294743242366043634083 ----------------------------------------------------------------------- 891922c21d6-50805444469590908054102603466659709215532265831475946218489 ----------------------------------------------------------------------- 268058419397748970c20d7- ----------------------------------------------------------------------- 25295782810188363280113412449907313808948937799446470909853008107629710 ----------------------------------------------------------------------- 6996390c19d8-1246191986018801429410111584461570796652924916418314809761 ----------------------------------------------------------------------- 13946206823293206015c18d9+ ----------------------------------------------------------------------- 27103817047931091700398998327649507434422969497617723073903265534396459 ----------------------------------------------------------------------- 1922930c17d10-114486890174628047503905050019569362204467411273629733348 ----------------------------------------------------------------------- 5029403659004631817730c16d11- ----------------------------------------------------------------------- 49990866977345339091346677191824639543052240299020304855078020731116499 ----------------------------------------------------------------------- 42575580c15d12-33576650708505682581779871569247986618312001081810803509 ----------------------------------------------------------------------- 40072866298343560135995c14d13+ ----------------------------------------------------------------------- 83807785896590438759909938478596635216530101325294318684248270925595212 ----------------------------------------------------------------------- 09493400c13d14+13123793346551568196932214015165838817109881075812283238 ----------------------------------------------------------------------- 332653386108502537742950c12d15+ ----------------------------------------------------------------------- 13508976413827284071339035218450289450916224719480912291059301569808179 ----------------------------------------------------------------------- 72054800c11d16-52826929822573747983905402289917359484276498763487170216 ----------------------------------------------------------------------- 84533290883998548133075c10d17+ ----------------------------------------------------------------------- 19401595270609005897588461661617962846090830639248297141449337400048609 ----------------------------------------------------------------------- 92973100c9d18+413624264628741353464836008373412673301256515872344486117 ----------------------------------------------------------------------- 2846092857887114331850c8d19- ----------------------------------------------------------------------- 38319862143341918907149178596338095635716342442714901790945262384863638 ----------------------------------------------------------------------- 79478800c7d20-909053528459169785393608041580865334493281727259622229775 ----------------------------------------------------------------------- 5062615709623980155875c6d21- ----------------------------------------------------------------------- 62473230346966299319987866479083529781998772108013924655654695210554416 ----------------------------------------------------------------------- 86860750c5d22-195298534157932551432326743075760027969841543859932808588 ----------------------------------------------------------------------- 7742681395670436476750c4d23- ----------------------------------------------------------------------- 10197991549113593137628886686082277217595319753781077375033249824433145 ----------------------------------------------------------------------- 25903750c3d24-118308066298032234221723410047845000450275082119865556665 ----------------------------------------------------------------------- 0353252518624537208500c2d25- ----------------------------------------------------------------------- 78520956960925223913969696887121797852828637378382200163973220935800676 ----------------------------------------------------------------------- 1356250cd26-25422454648905150135468774272202249969073704540820198082726 ----------------------------------------------------------------------- 4046943042234433750d27 | 1 1 o16 : Matrix S <--- S |