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 0x2a34540 -- [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 = 4187 -- ncalls = 10 -- nloop = 29 -- nsaved = 0 -- -- used 0.010998 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 0x215a600 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 4 at 0x2a34380 -- [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 0x2a341c0 -- [gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)m removing gb 1 at 0x2a34700 ooo{8}(2)oo -- number of (nonminimal) gb elements = 11 -- number of monomials = 267 -- ncalls = 10 -- nloop = 20 -- nsaved = 0 -- -- used 0.031995 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 0x215a300 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 7 3 1 2 5 2 3 1 2 9 6 2 7 2 10 o12 = ideal (-a + -a b + -a*b + b + -a c + -a*b*c + -b c + -a d + --a*b*d 6 2 3 3 5 7 5 9 ----------------------------------------------------------------------- 2 9 2 1 2 6 2 2 1 2 3 1 2 + 2b d + -a*c + -b*c + -a*c*d + 2b*c*d + -a*d + -b*d + c + -c d + 5 2 7 5 2 8 ----------------------------------------------------------------------- 8 2 1 3 3 3 2 8 2 3 2 3 2 1 2 -c*d + -d , 4a + -a b + -a*b + 2b + 2a c + -a*b*c + 4b c + -a d + 5 2 5 9 2 2 ----------------------------------------------------------------------- 2 1 2 2 2 3 5 1 2 3 2 4 3 -a*b*d + -b d + 2a*c + 3b*c + -a*c*d + -b*c*d + -a*d + --b*d + -c 5 2 4 7 3 10 3 ----------------------------------------------------------------------- 9 2 4 2 7 3 3 3 5 2 2 2 7 3 7 2 5 + --c d + -c*d + --d , --a + -a b + -a*b + -b + --a c + -a*b*c + 10 9 10 10 4 5 4 10 4 ----------------------------------------------------------------------- 2 2 10 5 2 6 2 7 2 3 5 8 2 b c + a d + --a*b*d + -b d + -a*c + -b*c + -a*c*d + -b*c*d + -a*d + 9 8 7 3 4 6 5 ----------------------------------------------------------------------- 2 2 3 2 2 1 3 -b*d + 5c + c d + c*d + -d ) 7 8 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 6 at 0x2a34e00 -- [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 0.13198 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 228176594864827313524978572769797555273475699962871837686290465031192 ----------------------------------------------------------------------- 576000000c27+3827986569020983554795728675708872122934345059948798168649 ----------------------------------------------------------------------- 57860009305702400000c26d- ----------------------------------------------------------------------- 20290506750514376781862876375907486140751274433052682758449776853013574 ----------------------------------------------------------------------- 45120000c25d2-137143282513457417077009891887018972541560381849785238244 ----------------------------------------------------------------------- 3971698098774794240000c24d3+ ----------------------------------------------------------------------- 26686529476158481968719174812997358902826280040804422327693303661274689 ----------------------------------------------------------------------- 97427200c23d4+383125859452247668665122579481420478584351414137113462031 ----------------------------------------------------------------------- 1180971114146762536960c22d5- ----------------------------------------------------------------------- 20835397179681066666822553153076487233283741803814209745525055273293391 ----------------------------------------------------------------------- 01598720c21d6-648052612447883387705416704894789917841071515717959083800 ----------------------------------------------------------------------- 7311091105164496186880c20d7- ----------------------------------------------------------------------- 19331244078769992464783181669900234528173982017569647240856814146811600 ----------------------------------------------------------------------- 13889920c19d8+340058465426690470456657815217837521289937393356180553392 ----------------------------------------------------------------------- 0352758032119090099640c18d9+ ----------------------------------------------------------------------- 23721359952148103864264087640954736224146946754722013586898521928618988 ----------------------------------------------------------------------- 84205592c17d10-35450090387460976958149298690875223430426018196587869415 ----------------------------------------------------------------------- 2002566194366805030168c16d11- ----------------------------------------------------------------------- 19348442715336319561168872310175247775332789113003842611776798032884507 ----------------------------------------------------------------------- 84759252c15d12-19933552318229164158394524397301176164477928640100694842 ----------------------------------------------------------------------- 16098769574638029113748c14d13- ----------------------------------------------------------------------- 29863202677621682206883852573415536477279741578625565252699933677810247 ----------------------------------------------------------------------- 8112540c13d14+173145436555835896741651537610429893756448639232344367859 ----------------------------------------------------------------------- 93710544385896368616c12d15- ----------------------------------------------------------------------- 48129374260801208399034043677746855488312253430154867985584164868651932 ----------------------------------------------------------------------- 8541134c11d16-240247971502834485416792975791627755754329240504971664233 ----------------------------------------------------------------------- 290242574823469623049c10d17- ----------------------------------------------------------------------- 35957267489663859936671355507216039348572841049986545931741016502396263 ----------------------------------------------------------------------- 493408c9d18-18399842962900756867859333567817357343430213037790397149233 ----------------------------------------------------------------------- 49976798967363992c8d19+ ----------------------------------------------------------------------- 16427742083865141678330675853873697827355061619592878589868288854000084 ----------------------------------------------------------------------- 124960c7d20+87445035572515215874962443947854458514052239130937361975253 ----------------------------------------------------------------------- 76626585676948496c6d21- ----------------------------------------------------------------------- 26692613977013662537958058728129547541809363542469310728946440195451325 ----------------------------------------------------------------------- 55072c5d22-369815703919755578787354152001487147613814566699342764135870 ----------------------------------------------------------------------- 284146633975808c4d23+33826935519550733311670054098814314233416586278996 ----------------------------------------------------------------------- 92624000142297814106425600c3d24+ ----------------------------------------------------------------------- 93135253974962680637249856352808701993804178467483960126234532706600312 ----------------------------------------------------------------------- 3456c2d25-3507555703263293775341523301194184519841122879210166269481754 ----------------------------------------------------------------------- 69312953871360cd26-2013522665488634247559473472734802403216771381524597 ----------------------------------------------------------------------- 96664149793790609623040d27 | 1 1 o16 : Matrix S <--- S |