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 0x2513540 -- [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 = 4184 -- ncalls = 10 -- nloop = 29 -- nsaved = 0 -- -- used 0.03 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 0x2043f00 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 4 at 0x2513380 -- [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 0x25131c0 -- [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 0.01 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 0x2043b00 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 9 3 2 4 2 9 3 5 2 1 1 2 2 9 o12 = ideal (-a + a b + -a*b + -b + -a c + -a*b*c + -b c + 5a d + -a*b*d + 5 3 4 3 4 3 5 ----------------------------------------------------------------------- 2 7 2 2 2 2 2 2 2 3 2 2b d + -a*c + -b*c + a*c*d + 9b*c*d + -a*d + 3b*d + -c + 3c d + 4 3 7 5 ----------------------------------------------------------------------- 2 10 3 3 7 2 2 7 3 7 2 2 2 2 3c*d + --d , 2a + -a b + a*b + -b + -a c + 8a*b*c + 2b c + -a d + 7 8 3 9 7 ----------------------------------------------------------------------- 1 7 2 2 2 2 5 2 2 3 2 3 -a*b*d + -b d + 2a*c + -b*c + 2a*c*d + -b*c*d + -a*d + -b*d + 2c + 2 6 5 2 9 8 ----------------------------------------------------------------------- 2 2 5 2 3 10 3 5 2 2 3 10 2 9 1 2 -c d + -c*d + d , --a + -a b + a*b + 2b + --a c + -a*b*c + -b c + 9 6 9 4 3 5 7 ----------------------------------------------------------------------- 8 2 1 8 2 5 2 2 2 6 7 2 3 2 -a d + -a*b*d + -b d + -a*c + -b*c + -a*c*d + -b*c*d + 5a*d + --b*d 9 3 9 4 3 7 5 10 ----------------------------------------------------------------------- 2 3 4 2 2 3 3 + -c + -c d + c*d + -d ) 3 9 2 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 6 at 0x2513000 -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mm removing gb 1 at 0x2513700 m{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 0.22 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 265527394232862042300173147657999909870145339168477030013655253015305 ----------------------------------------------------------------------- 69533440c27+48217463866416851341429136531419138826136921985197197441096 ----------------------------------------------------------------------- 8901829295614732288c26d+ ----------------------------------------------------------------------- 16103791549752475106857567706176516525551860673547369441490523358387208 ----------------------------------------------------------------------- 29582848c25d2+702880296686632350657349955667676786552635852761365666749 ----------------------------------------------------------------------- 53378365036419550150656c24d3+ ----------------------------------------------------------------------- 12195233033237271147154251338991459894435559427303886007543906928753545 ----------------------------------------------------------------------- 65338910528c23d4-123427021106068174133873967589587486549169662747353576 ----------------------------------------------------------------------- 66362179800502764384801426688c22d5- ----------------------------------------------------------------------- 14741387792127015421366732665352164857208555981423215127905721219534241 ----------------------------------------------------------------------- 052383333944c21d6+20244602217651855445270101798491119398835093587284475 ----------------------------------------------------------------------- 644941738715569734788551812198c20d7+ ----------------------------------------------------------------------- 55531200938454547475949947107287336431436763253716069470478340631498881 ----------------------------------------------------------------------- 191407859704c19d8+10472406969570480027736282656691231798604646927155024 ----------------------------------------------------------------------- 407388270109539677631770124739c18d9- ----------------------------------------------------------------------- 45781351251985629903147031001244414707505906937646082230044006070361083 ----------------------------------------------------------------------- 414887609536c17d10-1000272101073336246266816327233456107397470224256121 ----------------------------------------------------------------------- 59252163492254140733462461596036c16d11+ ----------------------------------------------------------------------- 14804848243808544768617016074773508184074061588118301548887907710783477 ----------------------------------------------------------------------- 0937355513188c15d12+138914596920905899597233940774678997853775654740843 ----------------------------------------------------------------------- 161291625857839065975241493514136c14d13- ----------------------------------------------------------------------- 44431501995532315722508782657125742820443171477083844288617377488502123 ----------------------------------------------------------------------- 297284498904c13d14+9767276643688653734935725966196021529016656827515956 ----------------------------------------------------------------------- 8702593428592593196115408421525c12d15+ ----------------------------------------------------------------------- 13598712504064688530713926762020067449618559678571756622672631814733668 ----------------------------------------------------------------------- 2015873654256c11d16+287511047590001903159713993276429528497807549787499 ----------------------------------------------------------------------- 48717646053356012332328472881687c10d17+ ----------------------------------------------------------------------- 26938840417437792140564823920947065920031706019786411870827453110618239 ----------------------------------------------------------------------- 526417418616c9d18+42571817092363842259096377157532098773380708746913783 ----------------------------------------------------------------------- 728496609995120538339514501460c8d19+ ----------------------------------------------------------------------- 24350573998639706831435675494398581718246307892062528068452631328408390 ----------------------------------------------------------------------- 518329675000c7d20+16042627594930313583873354180645018203102651795627871 ----------------------------------------------------------------------- 070029550265580082164194577816c6d21+ ----------------------------------------------------------------------- 12649867251777747455304890422936229784418441759113346250056041908822494 ----------------------------------------------------------------------- 442542322880c5d22+62256297052372816307132250547482219955045238318334768 ----------------------------------------------------------------------- 70165635546072274405004560080c4d23+ ----------------------------------------------------------------------- 25975435643742306342366111398389629348020665255376515996345782729460234 ----------------------------------------------------------------------- 33920242880c3d24+113343426523509751260009046044426386764282851171450109 ----------------------------------------------------------------------- 5662194592763091179175801920c2d25+ ----------------------------------------------------------------------- 29517352205537103813541971556474987564354424253842655286414316136899484 ----------------------------------------------------------------------- 6534447360cd26+26413867869825313300343586364826250590288897970455649508 ----------------------------------------------------------------------- 439889811926183776156160d27 | 1 1 o16 : Matrix S <--- S |