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 5.282e-6 seconds 3 6 9 o6 = 1 - 3T + 3T - T o6 : ZZ[T] |
i7 : time gens gb I; -- registering gb 2 at 0x44c7700 -- [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 = 29 -- nsaved = 0 -- -- used 0.0109068 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 0x4a73c30 |
i9 : I = ideal random(R^1, R^{3:-3}); -- registering gb 3 at 0x44c7540 -- [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 0x44c7380 -- [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 0.005664 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 0x4638f00 o11 = S o11 : PolynomialRing |
i12 : J = substitute(I,S) 3 3 4 2 2 3 3 10 2 1 1 2 7 2 o12 = ideal (-a + -a b + 2a*b + -b + --a c + -a*b*c + -b c + --a d + 8 9 4 9 2 9 10 ----------------------------------------------------------------------- 5 1 2 5 2 2 4 10 10 2 7 2 -a*b*d + -b d + -a*c + 5b*c + -a*c*d + --b*c*d + --a*d + -b*d + 3 3 3 5 3 9 4 ----------------------------------------------------------------------- 3 3 1 2 3 2 7 3 3 1 2 3 2 7 3 9 2 1 --c + -c d + -c*d + -d , a + -a b + -a*b + -b + -a c + -a*b*c + 10 6 4 4 3 7 2 7 5 ----------------------------------------------------------------------- 8 2 2 1 1 2 4 2 2 4 1 9 2 -b c + a d + -a*b*d + -b d + -a*c + 8b*c + -a*c*d + -b*c*d + --a*d + 3 9 4 7 9 8 10 ----------------------------------------------------------------------- 2 2 3 1 2 3 2 2 3 8 3 2 4 2 3 3 5 2 b*d + -c + -c d + -c*d + -d , -a + a b + -a*b + -b + -a c + 3 2 8 5 3 5 5 2 ----------------------------------------------------------------------- 9 2 1 2 1 2 2 2 1 2 7 1 6a*b*c + -b c + -a d + -a*b*d + 2b d + -a*c + -b*c + --a*c*d + -b*c*d 5 5 7 7 3 10 9 ----------------------------------------------------------------------- 2 2 3 1 2 5 2 3 + a*d + 3b*d + 2c + -c d + -c*d + 2d ) 5 7 o12 : Ideal of S |
i13 : installHilbertFunction(J, hf) |
i14 : gbTrace=3 o14 = 3 |
i15 : time gens gb J; -- registering gb 5 at 0x44c71c0 -- [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 --removing gb 0 at 0x44c78c0 -- {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 0.117442 seconds 1 39 o15 : Matrix S <--- S |
i16 : selectInSubring(1,gens gb J) o16 = | 177245913601111347880094022698723777431133014825406072306894839334446 ----------------------------------------------------------------------- 06365007872000c27+72411286982173778781592528353754559143814419791858402 ----------------------------------------------------------------------- 721066866242294559785091072000c26d+ ----------------------------------------------------------------------- 16439304899386640216458572071092390008060072913518009824121098795028593 ----------------------------------------------------------------------- 4330327859200c25d2+1932167631410376426416373639816637686364885463943858 ----------------------------------------------------------------------- 71032216861443314465841480499200c24d3+ ----------------------------------------------------------------------- 12103085001634882898661295104596015408506245286769769187149265967910987 ----------------------------------------------------------------------- 0384082083840c23d4+2973120972356467216254020921346313920308385747860001 ----------------------------------------------------------------------- 9312529650844640978073475178496c22d5+ ----------------------------------------------------------------------- 12847438520910108543728223801665809256330811588423784253808324254377254 ----------------------------------------------------------------------- 3749098053504c21d6+2992527053969806956659760812933454820017369014881653 ----------------------------------------------------------------------- 41266851637931682351242355938688c20d7+ ----------------------------------------------------------------------- 26534242783224743750608892035590295963923630135274929088447371961582928 ----------------------------------------------------------------------- 2141256631552c19d8-1141974737848236096396241105822753905588611178158662 ----------------------------------------------------------------------- 55264169003268001182559815476456c18d9- ----------------------------------------------------------------------- 46635445001363146320221354219088764088964149458629849586274695076320460 ----------------------------------------------------------------------- 1387711573320c17d10-435049804114009607607366426889087440765189729026804 ----------------------------------------------------------------------- 480953060037324088532067061571704c16d11- ----------------------------------------------------------------------- 56332403580172280110042739487361215492043154614287659982144797806422042 ----------------------------------------------------------------------- 166732561986c15d12+2591594642727219789027796325550336279297500432725351 ----------------------------------------------------------------------- 80508711203994917890687439904884c14d13+ ----------------------------------------------------------------------- 23482607575741821815165769173898080543691533542109902383106232735418773 ----------------------------------------------------------------------- 4980104912749c13d14+506508524829509745734143846815617512494369124775898 ----------------------------------------------------------------------- 43276921603731591816030150181571c12d15- ----------------------------------------------------------------------- 13081165008038348548483252090426813985952198763327168156749730185631950 ----------------------------------------------------------------------- 220937711258c11d16-4095602932259716984702914653847719875864851784869268 ----------------------------------------------------------------------- 9871387009718581097613262562864c10d17- ----------------------------------------------------------------------- 71085345865426273816148069356719562158547295439343264681583042555722635 ----------------------------------------------------------------------- 391720336488c9d18-49393810696081976681758915736861180690949472502650494 ----------------------------------------------------------------------- 573748750419551708627346020128c8d19- ----------------------------------------------------------------------- 12210183967020483652851954531059636067143828141768719537033101734737890 ----------------------------------------------------------------------- 572950072480c7d20-43687830929402193766015413874956284832939608487372927 ----------------------------------------------------------------------- 83001086644036559800024692800c6d21- ----------------------------------------------------------------------- 33601975616731552612974192077490054224630230558271420991181971151884464 ----------------------------------------------------------------------- 19532646400c5d22-506756537559314586419784525286729118903436159350073266 ----------------------------------------------------------------------- 572191463922375033713002240c4d23+ ----------------------------------------------------------------------- 22070368192259553004475209181571727221207668113038788652315751239101528 ----------------------------------------------------------------------- 6935180032c3d24+2900276152812688573875047537546621928411111338727157559 ----------------------------------------------------------------------- 13229268209901868579840c2d25- ----------------------------------------------------------------------- 30979626159850238343413004432218759097853639295328833503817217842524720 ----------------------------------------------------------------------- 786157568cd26-523188258539336242798680203672939116183921910269785122111 ----------------------------------------------------------------------- 4959326165939949543424d27 | 1 1 o16 : Matrix S <--- S |