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 9.892e-6 seconds 3 6 9 o6 = 1 - 3T + 3T - T o6 : ZZ[T] |
i7 : time gens gb I; -- registering gb 2 at 0x95ae130 -- [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 = 4183 -- ncalls = 10 -- nloop = 23 -- nsaved = 0 -- -- used 0.020854 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 0x840ea00 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 3 at 0x9204be0 -- [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 0x9204980 -- [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 0.017004 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 0x840e980 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 5 3 2 1 2 1 3 6 2 1 2 2 5 o12 = ideal (-a + 5a b + -a*b + -b + -a c + 3a*b*c + -b c + a d + -a*b*d + 9 2 2 5 3 8 ----------------------------------------------------------------------- 6 2 1 2 1 2 9 4 2 3 2 5 3 9 2 -b d + -a*c + -b*c + -a*c*d + 6b*c*d + -a*d + -b*d + -c + --c d + 5 2 2 7 5 4 3 10 ----------------------------------------------------------------------- 2 9 3 10 3 5 2 4 2 5 3 3 2 5 4 2 1 2 3c*d + --d , --a + -a b + -a*b + -b + -a c + -a*b*c + -b c + -a d + 10 9 6 3 2 5 4 5 4 ----------------------------------------------------------------------- 7 2 1 2 5 2 7 3 2 9 2 3 -a*b*d + 2b d + -a*c + -b*c + -a*c*d + b*c*d + -a*d + -b*d + c + 8 9 6 3 2 7 ----------------------------------------------------------------------- 2 2 3 2 2 3 1 3 2 5 2 8 3 1 2 2 -c d + --c*d + -d , -a + a b + -a*b + -b + -a c + 5a*b*c + 5b c + 5 10 5 4 8 9 7 ----------------------------------------------------------------------- 1 2 1 2 4 2 4 2 10 3 4 2 -a d + -a*b*d + 3b d + -a*c + -b*c + --a*c*d + --b*c*d + -a*d + 2 2 5 7 7 10 3 ----------------------------------------------------------------------- 2 1 3 2 2 2 9 3 3b*d + -c + 3c d + -c*d + -d ) 8 7 4 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 5 at 0x92045f0 -- [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 --removing gb 0 at 0x95ae000 {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 0.56568 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 264412176073022421979018634273551914813021113892630187414948920077138 ----------------------------------------------------------------------- 00107810816000c27+24596781394808868984650612155081958370995279977907478 ----------------------------------------------------------------------- 1795717533097064665468012953600c26d- ----------------------------------------------------------------------- 19421897099824739332586798286700976529221384691841898770433795438080911 ----------------------------------------------------------------------- 86275934865408c25d2+539222308669192956709904062575726675922745170637660 ----------------------------------------------------------------------- 3349046614643372607335981173285888c24d3- ----------------------------------------------------------------------- 67334923109695168263304006952454171476335558336907760464785638757429062 ----------------------------------------------------------------------- 32911783649280c23d4+139117500115245078133863080330788583874526964293814 ----------------------------------------------------------------------- 53499977302509382744466194434105088c22d5- ----------------------------------------------------------------------- 50875433185683161431481453100015175820646698216260634598493836564482021 ----------------------------------------------------------------------- 580587465344512c21d6+13768643282206758929753088628510515629245173795611 ----------------------------------------------------------------------- 3815782000871056406089672530640182592c20d7- ----------------------------------------------------------------------- 23537822734334713487782378785367076985955842534122919956906446886793563 ----------------------------------------------------------------------- 2277152066959024c19d8+3193056791806501700434896081933873718785024558532 ----------------------------------------------------------------------- 66516209266851075584963855283134835392c18d9- ----------------------------------------------------------------------- 45224228563732414918001614265809348081484760820807983385050104928149376 ----------------------------------------------------------------------- 2140250545703232c17d10+ ----------------------------------------------------------------------- 40321191799439867889127971754657232080595939471760194968766020975057651 ----------------------------------------------------------------------- 2475909032538176c16d11- ----------------------------------------------------------------------- 45278489922553823381013188989350383417301779141018005547923113880141401 ----------------------------------------------------------------------- 0738705521971728c15d12+ ----------------------------------------------------------------------- 45545634007625830781044175700783268890753130529326188274657393561281006 ----------------------------------------------------------------------- 4176593024006328c14d13- ----------------------------------------------------------------------- 22275107404842171301035733922986199800699834570869998516178237188318146 ----------------------------------------------------------------------- 7700518252570256c13d14+ ----------------------------------------------------------------------- 43933071801476466492722397460232698443287292509222536479293670462852319 ----------------------------------------------------------------------- 8702115892330208c12d15- ----------------------------------------------------------------------- 34255877961708569660669464978848627073288887375241808991722442745320470 ----------------------------------------------------------------------- 312208047738532c11d16+2787934391751104952750356187319235383987316359075 ----------------------------------------------------------------------- 02775540659040738616113060403497336896c10d17+ ----------------------------------------------------------------------- 22685711301828342967710704629156197885361130258612657490116502528696129 ----------------------------------------------------------------------- 7311750079089c9d18+9974016737239954042798601574662556436143585590197799 ----------------------------------------------------------------------- 5234398640652009107161910742466840c8d19- ----------------------------------------------------------------------- 11698155697247430491651383136645170876140271460812598736730924155812408 ----------------------------------------------------------------------- 532411974212680c7d20+18146598441125799472777338699704984904371343762980 ----------------------------------------------------------------------- 697955259147415467417414280108734550c6d21- ----------------------------------------------------------------------- 63010375313057808448927003520808307390261693059670837195593886508092872 ----------------------------------------------------------------------- 87145583866800c5d22+213082810715481183531729758651257738495430917870449 ----------------------------------------------------------------------- 0407454069154490390355836821512000c4d23- ----------------------------------------------------------------------- 87026404778373079070194824375324661396677012089014996244587815857363097 ----------------------------------------------------------------------- 8680696276500c3d24+2493613214229794395130430589476898484885077796350107 ----------------------------------------------------------------------- 14133675007450164263789769060000c2d25- ----------------------------------------------------------------------- 18846752508030919986608453891944953388574789925405882545975954639355515 ----------------------------------------------------------------------- 790139740000cd26+102132037881237880111969833922015053526301181005551647 ----------------------------------------------------------------------- 0439851465239604692787285000d27 | 1 1 o16 : Matrix S <--- S |