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 0.00001718 seconds 3 6 9 o6 = 1 - 3T + 3T - T o6 : ZZ[T] |
i7 : time gens gb I; -- registering gb 2 at 0x3fc7000 -- [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 -- #reduction steps = 38 -- #spairs done = 11 -- ncalls = 10 -- nloop = 29 -- nsaved = 0 -- -- used 0.0322393 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 0x4f0e960 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 3 at 0x200de00 -- [gb]number of (nonminimal) gb elements = 0 -- number of monomials = 0 -- #reduction steps = 0 -- #spairs done = 0 -- ncalls = 0 -- nloop = 0 -- nsaved = 0 -- o9 : Ideal of R |
i10 : time hf = poincare I -- registering gb 4 at 0x200dc40 -- [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 -- #reduction steps = 236 -- #spairs done = 30 -- ncalls = 10 -- nloop = 20 -- nsaved = 0 -- -- used 0.014268 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 0x4f0e870 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 4 3 5 2 2 2 5 3 2 2 2 3 o12 = ideal (-a + -a b + -a*b + -b + 9a c + 2a*b*c + 3b c + a d + -a*b*d + 9 6 9 7 4 ----------------------------------------------------------------------- 2 9 2 3 2 2 9 2 2 3 2 8 3 1 2 3b d + --a*c + -b*c + -a*c*d + -b*c*d + -a*d + -b*d + -c + -c d + 10 5 3 8 7 4 7 4 ----------------------------------------------------------------------- 7 2 2 3 1 3 5 2 5 2 8 3 1 2 1 2 9 2 -c*d + -d , -a + -a b + -a*b + -b + --a c + -a*b*c + 3b c + --a d + 3 5 5 9 8 3 10 3 10 ----------------------------------------------------------------------- 2 10 2 2 2 3 3 2 1 2 3 3 a*b*d + 3b d + --a*c + -b*c + -a*c*d + -b*c*d + 6a*d + -b*d + -c + 3 3 2 5 2 2 ----------------------------------------------------------------------- 1 2 1 2 1 3 4 3 3 2 2 3 2 10 8 2 -c d + -c*d + -d , -a + -a b + 6a*b + b + 7a c + --a*b*c + -b c + 2 2 9 3 5 7 5 ----------------------------------------------------------------------- 2 1 2 2 5 2 2 2 5 5 2 2 5 2 2a d + -a*b*d + -b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d + -b*d 9 3 3 5 9 2 3 6 ----------------------------------------------------------------------- 4 3 2 2 5 3 + -c + 9c d + 2c*d + -d ) 9 3 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 5 at 0x200da80 -- [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) --removing gb 0 at 0x3fc71c0 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 -- #reduction steps = 284 -- #spairs done = 53 -- ncalls = 46 -- nloop = 54 -- nsaved = 0 -- -- used 0.1857 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 121104465040086803108721097863108870792683988771548964278539418771772 ----------------------------------------------------------------------- 991850086400c27-3674838465595108895167862631543139723635401490767438027 ----------------------------------------------------------------------- 120676292289552300550144000c26d+ ----------------------------------------------------------------------- 28120209593004408645878086154163126632271503172583168830927990013457456 ----------------------------------------------------------------------- 0266568417280c25d2-5839650123583764078458584668685101622613185073770807 ----------------------------------------------------------------------- 71513829519292706035096955613440c24d3+ ----------------------------------------------------------------------- 49297713052930750797436833283285837021175236066108979750891542064448973 ----------------------------------------------------------------------- 120746722121600c23d4+19576540112496027125389757990723941375228675384901 ----------------------------------------------------------------------- 4136330292249816755256346687768460260c22d5+ ----------------------------------------------------------------------- 60410556266936436420899128343680845294760965522829622827101885707599138 ----------------------------------------------------------------------- 3889952081727980c21d6+1236926208892918201847086543016972216357556504114 ----------------------------------------------------------------------- 958834324794221828350968278689614142035c20d7+ ----------------------------------------------------------------------- 21194389297011101847190739123363143766198667417365386586571894155948044 ----------------------------------------------------------------------- 10519265929228367c19d8+ ----------------------------------------------------------------------- 29817445532020895870429152562792696436352758583868057607805993616835261 ----------------------------------------------------------------------- 25312764882086640c18d9+ ----------------------------------------------------------------------- 35895618278948883546466312042461245129087281662149066268714727949564831 ----------------------------------------------------------------------- 26412968975622265c17d10+ ----------------------------------------------------------------------- 37128626112551948809140213204427978308247279890913481183816397052803478 ----------------------------------------------------------------------- 38260606732091162c16d11+ ----------------------------------------------------------------------- 33461763032863955989533252907486909290664270457850439161834773645140806 ----------------------------------------------------------------------- 68990753928903754c15d12+ ----------------------------------------------------------------------- 26641908402491945940827157015663353363636837491919617409584270441873262 ----------------------------------------------------------------------- 94714134783733135c14d13+ ----------------------------------------------------------------------- 18916719416081365427888278380914378343514114790985385622484772183332467 ----------------------------------------------------------------------- 64089548006650936c13d14+ ----------------------------------------------------------------------- 12056182152276117943381920880853826641525595129378737047002442468550028 ----------------------------------------------------------------------- 52483955714921443c12d15+ ----------------------------------------------------------------------- 69813787064512476903410839812770359136193514501046808502025650419992180 ----------------------------------------------------------------------- 3536861782871188c11d16+ ----------------------------------------------------------------------- 36768186253065869061760372756149885736300833186957896877259921736393158 ----------------------------------------------------------------------- 2359880465237216c10d17+ ----------------------------------------------------------------------- 17676812172178853125673786646700805838168371983111457171456643945828750 ----------------------------------------------------------------------- 6195108116864236c9d18+7750132554533623476146001119530771699348122829438 ----------------------------------------------------------------------- 5405397451230923749438746921544834280c8d19+ ----------------------------------------------------------------------- 30716784937515487226190924442502891869086978712453985255605590639511277 ----------------------------------------------------------------------- 558927204174760c7d20+10972308005134773833643382525979845723302725650487 ----------------------------------------------------------------------- 244643743717259305939317407706378640c6d21+ ----------------------------------------------------------------------- 34673909701654583355070689410880090586020308057134757094399261862352611 ----------------------------------------------------------------------- 06494584618400c5d22+967235323166925145489660053743041448278183528212996 ----------------------------------------------------------------------- 029843306729880776009598831027360c4d23+ ----------------------------------------------------------------------- 22734788341616907264733403881679931270923088778071616630599589486042054 ----------------------------------------------------------------------- 1780976617280c3d24+4605023755453171286540840006331077655060401234026243 ----------------------------------------------------------------------- 4188623626550878760720344772480c2d25+ ----------------------------------------------------------------------- 65797861500504314940346553952118021162574456361270253032510085287808113 ----------------------------------------------------------------------- 26208817920cd26+8088882660207702690504428865193691359249673321712264213 ----------------------------------------------------------------------- 64792269049697989474913280d27 | 1 1 o16 : Matrix S <--- S |