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 0x25fe380 -- [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 = 4177 -- ncalls = 10 -- nloop = 29 -- nsaved = 0 -- -- used 0.020001 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 0x24ab800 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 4 at 0x25fe1c0 -- [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 0x25fe000 -- [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. 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 0x24ab500 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 7 3 1 2 9 2 7 3 10 2 3 2 9 2 o12 = ideal (-a + --a b + -a*b + --b + --a c + -a*b*c + 10b c + --a d + 4 10 4 10 3 8 10 ----------------------------------------------------------------------- 8 3 2 1 2 1 2 7 2 5 2 1 3 -a*b*d + -b d + -a*c + -b*c + a*c*d + b*c*d + --a*d + -b*d + -c + 9 2 2 6 10 4 4 ----------------------------------------------------------------------- 6 2 2 10 3 5 3 5 2 1 2 5 3 9 2 1 2 -c d + 7c*d + --d , -a + -a b + -a*b + -b + --a c + a*b*c + -b c + 5 3 7 6 3 6 10 3 ----------------------------------------------------------------------- 4 2 2 9 2 2 6 3 2 2 2 2 -a d + 2a*b*d + b d + -a*c + b*c + -a*c*d + -b*c*d + -a*d + -b*d + 3 5 5 8 5 3 ----------------------------------------------------------------------- 3 4 2 5 2 3 3 3 8 2 9 2 2 3 2 4 3c + -c d + -c*d + d , -a + -a b + -a*b + -b + 2a c + -a*b*c + 5 2 4 9 8 5 5 ----------------------------------------------------------------------- 8 2 2 7 4 2 8 2 3 2 2 1 1 2 -b c + 6a d + -a*b*d + -b d + -a*c + --b*c + -a*c*d + -b*c*d + -a*d 3 2 5 3 10 3 2 5 ----------------------------------------------------------------------- 4 2 10 3 1 2 1 2 2 3 + -b*d + --c + -c d + -c*d + -d ) 5 7 4 5 7 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 6 at 0x2a96e00 -- [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 removing gb 1 at 0x25fe540 -- 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 0.220013 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 496459565544627592810347602320230085374233009745649341731045840933829 ----------------------------------------------------------------------- 276057600c27-3573162422522272740799431538399336115733239357063780571835 ----------------------------------------------------------------------- 542850774896683827200c26d- ----------------------------------------------------------------------- 87732354541810266558402797017499076507130980176846298359164908890878416 ----------------------------------------------------------------------- 163620864c25d2+70700089585345348479750430349880249480130089503441285539 ----------------------------------------------------------------------- 7705902038707622048832512c24d3+ ----------------------------------------------------------------------- 49230548433189373560536255583127181073384547375095195300676047651017536 ----------------------------------------------------------------------- 41308396160c23d4+346538917066943520487262962943583335203915732993858028 ----------------------------------------------------------------------- 29212109621453372148278585728c22d5+ ----------------------------------------------------------------------- 16864912840373810842373338903717562149381277530074919383059679398327927 ----------------------------------------------------------------------- 7795765152784c21d6+5550959552432522270711014662426970002652683457887375 ----------------------------------------------------------------------- 35427480345492478219092043340664c20d7+ ----------------------------------------------------------------------- 12918957837761518191737431326876985288954402585399848126345572848999003 ----------------------------------------------------------------------- 38191110312128c19d8+263104711646443743987238184981037213174836341170044 ----------------------------------------------------------------------- 9352735165970677194983590598544757c18d9+ ----------------------------------------------------------------------- 41826164465019076473570414459577661122329950255660546093426302154694048 ----------------------------------------------------------------------- 51759682823403c17d10+60503486352838425826597830108436209318796877397825 ----------------------------------------------------------------------- 71861889090675166341175760969478231c16d11+ ----------------------------------------------------------------------- 72785952481220640331702563709550893471796003510656199560966495210832893 ----------------------------------------------------------------------- 06656317605641c15d12+79251119819916162750764473157100390207605382762927 ----------------------------------------------------------------------- 61781633678549890287400497365973132c14d13+ ----------------------------------------------------------------------- 74843770336652367604688118063954185028202385543311914092985758446261105 ----------------------------------------------------------------------- 62693000253476c13d14+62687103784263982121730448813223606097326594748660 ----------------------------------------------------------------------- 74997477979278074972909156895274800c12d15+ ----------------------------------------------------------------------- 46571743838346883405850542337799986053355289549911537002870980689547683 ----------------------------------------------------------------------- 01125344706328c11d16+30040699695410945374835793310355024571943732481550 ----------------------------------------------------------------------- 25177318295656709730050276702630208c10d17+ ----------------------------------------------------------------------- 17091814645505446092579806565079465722002808627156834735067337983275567 ----------------------------------------------------------------------- 38638782928096c9d18+836653698753894112146307692457137356238722869118644 ----------------------------------------------------------------------- 848897735561196401309447769441024c8d19+ ----------------------------------------------------------------------- 34172839196461419839374538740906379568276516747976666005175349406458666 ----------------------------------------------------------------------- 8707277251840c7d20+1205180208822931428877698626539786758827424571388545 ----------------------------------------------------------------------- 66078344238189590005818428014080c6d21+ ----------------------------------------------------------------------- 37124381730923916387507274108805456231166459107886132566491034900215089 ----------------------------------------------------------------------- 394647626752c5d22+91869394280744352116755312049460419614603826689827589 ----------------------------------------------------------------------- 13371597309176895982027507712c4d23+ ----------------------------------------------------------------------- 16878967229107194049335236638306684814385814810215342325486359140999839 ----------------------------------------------------------------------- 63416002560c3d24+228122625612331900326446171094312288616314907997752460 ----------------------------------------------------------------------- 596935841948174766994227200c2d25+ ----------------------------------------------------------------------- 20956334376132969207416756444674086188647524374695854670554321530386452 ----------------------------------------------------------------------- 421017600cd26+808102999129241145309506363218010322271305519011239285825 ----------------------------------------------------------------------- 595349262571588812800d27 | 1 1 o16 : Matrix S <--- S |