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 0.000010757 seconds 3 6 9 o6 = 1 - 3T + 3T - T o6 : ZZ[T] |
i7 : time gens gb I; -- registering gb 2 at 0x24f5a80 -- [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 -- --removing gb 0 at 0x24f5c40 -- used 0.0479175 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 3 at 0x29b7f00 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 3 at 0x24f58c0 -- [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 4 at 0x24f5700 -- [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 -- ncalls = 10 -- nloop = 20 -- nsaved = 0 -- -- used 0.0080549 seconds 3 6 9 o10 = 1 - 3T + 3T - T o10 : ZZ[T] |
i11 : S = QQ[a..d,MonomialOrder=>Eliminate 2] -- registering polynomial ring 4 at 0x29b7e10 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 1 3 1 2 9 2 3 1 2 1 2 1 2 5 o12 = ideal (-a + -a b + --a*b + b + -a c + -a*b*c + b c + -a d + -a*b*d + 3 2 10 2 7 8 4 ----------------------------------------------------------------------- 2 2 7 2 2 5 4 7 2 4 2 4 3 7 2 -b d + -a*c + b*c + -a*c*d + -b*c*d + -a*d + -b*d + -c + --c d + 3 8 3 7 9 7 5 10 ----------------------------------------------------------------------- 2 2 8 3 1 3 2 3 2 5 3 5 2 8 4 2 1 2 -c*d + -d , --a + 3a b + -a*b + -b + -a c + -a*b*c + -b c + -a d + 5 7 10 8 8 4 3 3 3 ----------------------------------------------------------------------- 1 2 2 1 2 4 10 1 2 2 10 3 a*b*d + -b d + a*c + -b*c + -a*c*d + --b*c*d + --a*d + b*d + --c + 3 5 3 7 10 7 ----------------------------------------------------------------------- 4 2 2 4 3 1 3 4 2 2 3 9 2 9 8 2 -c d + 2c*d + -d , -a + -a b + 2a*b + 10b + -a c + --a*b*c + -b c + 5 5 2 3 4 10 3 ----------------------------------------------------------------------- 8 2 7 2 1 2 1 2 6 3 2 1 2 -a d + a*b*d + -b d + --a*c + -b*c + -a*c*d + -b*c*d + 2a*d + -b*d 3 4 10 2 5 5 3 ----------------------------------------------------------------------- 1 3 1 2 1 2 1 3 + -c + --c d + -c*d + -d ) 2 10 6 2 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 5 at 0x24f5540 -- [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 0.131304 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 414287576029981165469185823351031505651386214944793116045859905465929 ----------------------------------------------------------------------- 0712c27+105459986840532852388357090482855322476051303667667075852154116 ----------------------------------------------------------------------- 92667732508c26d-4169001831463835256357567953063095928167098238644280893 ----------------------------------------------------------------------- 0510283438622750598c25d2- ----------------------------------------------------------------------- 19635455285100534935173450266264731074546033267816600140643956278927272 ----------------------------------------------------------------------- 7331c24d3-1652999732602271737969895130385693592577260980520726111678733 ----------------------------------------------------------------------- 76347373264764c23d4+105758766384704288073456374700176260689562882538880 ----------------------------------------------------------------------- 6845417616064527965035431c22d5+ ----------------------------------------------------------------------- 63136087503625735475469229074924521352513639667065829829066400767228351 ----------------------------------------------------------------------- 24706c21d6+214621806957252778376080704044941590142201355827328328079034 ----------------------------------------------------------------------- 41093484571056916c20d7+ ----------------------------------------------------------------------- 53428439963520866130917955001743612838828265231827522135632363058032388 ----------------------------------------------------------------------- 150254c19d8+10668736569853458115721890540584004899651322036698969460910 ----------------------------------------------------------------------- 1754454477502863202c18d9+ ----------------------------------------------------------------------- 18049291835135293692718841080122342961412007367956773363794875060584816 ----------------------------------------------------------------------- 6117436c17d10+264643846581228499494692724971374938338612322683445997478 ----------------------------------------------------------------------- 011705795975307934399c16d11+ ----------------------------------------------------------------------- 34034987344566511279017307933568387402433972604148897376487357884669900 ----------------------------------------------------------------------- 0004050c15d12+389123184780999367073407355345975559048730066898320830364 ----------------------------------------------------------------------- 293200272147952353473c14d13+ ----------------------------------------------------------------------- 39908207023273256114424879778085299703571900581338795201753320348298548 ----------------------------------------------------------------------- 4213644c13d14+368591420192687906319158485366608903489736177603251308619 ----------------------------------------------------------------------- 300978575066276869188c12d15+ ----------------------------------------------------------------------- 30788395287002952873675966350221093774161596111424765464755300702083184 ----------------------------------------------------------------------- 1645820c11d16+233187463216909355773362460883718045264000314864267273778 ----------------------------------------------------------------------- 998872192733439996642c10d17+ ----------------------------------------------------------------------- 15971009577285563938859170289839611162403025075871237666598175245381526 ----------------------------------------------------------------------- 4714236c9d18+9858739398189604538553988814072269688400198862612211059247 ----------------------------------------------------------------------- 9309082889873002840c8d19+ ----------------------------------------------------------------------- 54640965424296866160396564460426411904413810381858260818795734911782644 ----------------------------------------------------------------------- 363160c7d20+26738369862008808632312927991268814714470039095970639802040 ----------------------------------------------------------------------- 096919411802983400c6d21+ ----------------------------------------------------------------------- 11418928976508273927408154376875554476318071243341710526056135778859072 ----------------------------------------------------------------------- 883000c5d22+41637237592269707439115954753165859317671586088373247346724 ----------------------------------------------------------------------- 06104922466947500c4d23+ ----------------------------------------------------------------------- 11913094792993503556338835821198261817392362888186892348332871135776130 ----------------------------------------------------------------------- 00000c3d24+262204016663935614828464991312308336895557252212476239368983 ----------------------------------------------------------------------- 465087492875000c2d25+23908085950417044000999220215553232781567757520284 ----------------------------------------------------------------------- 558123372058409791250000cd26- ----------------------------------------------------------------------- 10002021671218013487568778413335979595060218973608863342861760746234375 ----------------------------------------------------------------------- 000d27 | 1 1 o16 : Matrix S <--- S |