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installHilbertFunction -- install a Hilbert function without computation

Synopsis

Description

If M is a module, then hf should be the poincare polynomial of M. If M is an ideal, then hf should be the poincare polynomial of comodule M. If M is a matrix, then hf should be the poincare polynomial of cokernel M.

An installed Hilbert function will be used by Groebner 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 0x600000000009ca80

   -- [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                = 4193
   --      -- used 0.042 seconds

             1       11
o7 : Matrix R  <--- R
In this case, the savings is minimal, but often it can be dramatic.

Another important situation is to compute a Groebner basis using a different monomial order. In the example below

i8 : R = QQ[a..d];
i9 : I = ideal random(R^1, R^{3:-3});

   -- registering gb 4 at 0x600000000009c8c0

   -- [gb]number of (nonminimal) gb elements = 0
   -- number of monomials                = 0
   -- 
o9 : Ideal of R
i10 : time hf = poincare I

   -- registering gb 5 at 0x600000000009c700

   -- [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
   --      -- used 0.022 seconds

            3     6    9
o10 = 1 - 3T  + 3T  - T

o10 : ZZ [T]
i11 : S = QQ[a..d,MonomialOrder=>Eliminate 2]

o11 = S

o11 : PolynomialRing
i12 : J = substitute(I,S)

             5  3   9  2    7    2   5  3   2  2    6           2    7  2   
o12 = ideal (-*a  + -*a b + -*a*b  + -*b  + -*a c + -*a*b*c + 8b c + -*a d +
             4      2       6        6      5       5                6      
      -----------------------------------------------------------------------
      3         6  2    1    2   9    2   10         3         7    2  
      -*a*b*d + -*b d + -*a*c  + -*b*c  + --*a*c*d + -*b*c*d + -*a*d  +
      4         7       4        8         3         4         4       
      -----------------------------------------------------------------------
      8    2   1  3   4  2     9    2   1  3   3     2    1    2   3  3  
      -*b*d  + -*c  + -*c d + --*c*d  + -*d , a  + 8a b + -*a*b  + -*b  +
      7        2      3       10        4                 9        2     
      -----------------------------------------------------------------------
      4  2             2  2      2    1         4  2    8    2   1    2  
      -*a c + 4a*b*c + -*b c + 4a d + -*a*b*d + -*b d + -*a*c  + -*b*c  +
      7                3              2         9       9        9       
      -----------------------------------------------------------------------
      2                  3    2      2   1  3   3  2    1    2   3  3  4  3  
      -*a*c*d + 8b*c*d + -*a*d  + b*d  + -*c  + -*c d + -*c*d  + -*d , -*a  +
      3                  4               7      5       9        7     3     
      -----------------------------------------------------------------------
      9  2    3    2   1  3   2  2    5         7  2    9  2    7        
      -*a b + -*a*b  + -*b  + -*a c + -*a*b*c + -*b c + -*a d + -*a*b*d +
      5       7        9      3       9         4       8       2        
      -----------------------------------------------------------------------
      4  2    1    2   9    2   5         9             2   2    2    1  3  
      -*b d + -*a*c  + -*b*c  + -*a*c*d + -*b*c*d + 2a*d  + -*b*d  + --*c  +
      5       3        4        7         7                 3        10     
      -----------------------------------------------------------------------
       2    7    2   7  3
      c d + -*c*d  + -*d )
            6        4

o12 : Ideal of S
i13 : installHilbertFunction(J, hf)
i14 : gbTrace=3

o14 = 3
i15 : time gens gb J;

   -- registering gb 6 at 0x600000000009c380

   -- [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 1 at 0x600000000009cc40
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
   --      -- used 0.421 seconds

              1       39
o15 : Matrix S  <--- S
i16 : selectInSubring(1,gens gb J)

   -- [gb]number of (nonminimal) gb elements = 39
   -- number of monomials                = 1051
   -- 
o16 = | 109951076075589603607379639092920107191077842047433285281607021558810
      -----------------------------------------------------------------------
      049883707801600000c27-4686924695125355562559234266999248755751350152530
      -----------------------------------------------------------------------
      37290637037940007124467934215536640000c26d-
      -----------------------------------------------------------------------
      12888291820625910834073001746119126860044705851357375766895168428954565
      -----------------------------------------------------------------------
      25580988547072000c25d2+
      -----------------------------------------------------------------------
      95311851768318039167855679599244940831952767216781272775431316971641562
      -----------------------------------------------------------------------
      02287755783372800c24d3-
      -----------------------------------------------------------------------
      19720663914619462529571958494452271162838828009511110238198796727099147
      -----------------------------------------------------------------------
      391747155520716800c23d4+
      -----------------------------------------------------------------------
      22631931494755499156618325247568305540815947458458538169339088046843149
      -----------------------------------------------------------------------
      170261940368793600c22d5+
      -----------------------------------------------------------------------
      79180770775424934311786219321813577090903288687488746822676303042658723
      -----------------------------------------------------------------------
      947151877874736640c21d6-
      -----------------------------------------------------------------------
      70852865029503702076931394367454782497013534306955737456314674042082334
      -----------------------------------------------------------------------
      5481124281311806720c20d7-
      -----------------------------------------------------------------------
      63056696006154486460476392098586257274854299612350762644665814420335589
      -----------------------------------------------------------------------
      5854949894261658304c19d8-
      -----------------------------------------------------------------------
      17663164893499106882017960061881991268977192484964368197064177275083427
      -----------------------------------------------------------------------
      60407072362578802280c18d9-
      -----------------------------------------------------------------------
      23706232902010981845080453180190524362336194827712777118643619757153629
      -----------------------------------------------------------------------
      94585345774342091240c17d10-
      -----------------------------------------------------------------------
      36248816708024852747912681911690706186309968240027100522620311660437244
      -----------------------------------------------------------------------
      46325461005320349600c16d11-
      -----------------------------------------------------------------------
      38863052051014026424587160206641671418633853757001941879862235459260343
      -----------------------------------------------------------------------
      49271813881604730150c15d12-
      -----------------------------------------------------------------------
      42959781501395150748401784193043644329064842704838213752292269175783430
      -----------------------------------------------------------------------
      4388677844758267500c14d13+
      -----------------------------------------------------------------------
      75742107456609238709616906536179732666715184557906163317363104456553863
      -----------------------------------------------------------------------
      59558924864900861250c13d14+
      -----------------------------------------------------------------------
      56893630198252618257292705320362082297010112147121791406470851826037924
      -----------------------------------------------------------------------
      39139722553157791875c12d15-
      -----------------------------------------------------------------------
      40438448888370763841981288825165492438896938677878008146699569009349293
      -----------------------------------------------------------------------
      36101128769251225000c11d16-
      -----------------------------------------------------------------------
      75486844549582972533333155477247890096429994635256520997565687641033889
      -----------------------------------------------------------------------
      60351992505345662500c10d17-
      -----------------------------------------------------------------------
      35055422859066880578033780619139560310564026128104277379786246946290189
      -----------------------------------------------------------------------
      17189504723840537500c9d18-
      -----------------------------------------------------------------------
      13105445664092883265927311090374317649773893375042661029784648290644147
      -----------------------------------------------------------------------
      77285136774884375000c8d19+
      -----------------------------------------------------------------------
      19735417636311826225733525052852991499590850140339844318804800179871016
      -----------------------------------------------------------------------
      9840400773925000000c7d20+
      -----------------------------------------------------------------------
      20427445564098785302443614268027449606508541419343696274910467558869112
      -----------------------------------------------------------------------
      755446428012500000c6d21-
      -----------------------------------------------------------------------
      25532316737975998333943559832484179655919920706895946123659828966358680
      -----------------------------------------------------------------------
      760982658312500000c5d22-
      -----------------------------------------------------------------------
      54476682784538556757997883635539857254735345641479894160372762788175869
      -----------------------------------------------------------------------
      85121553750000000c4d23-
      -----------------------------------------------------------------------
      31813421210247519349180671637350302853585189848821503364452698068174585
      -----------------------------------------------------------------------
      7383125000000000c3d24-5394190915687107576430451947341829022667216566107
      -----------------------------------------------------------------------
      343532021994385600075028750000000000c2d25+
      -----------------------------------------------------------------------
      11971077024379141772176603915187908057028816122601812328775518067594270
      -----------------------------------------------------------------------
      000000000000cd26-253515663350606316265600092747204231107338737527376715
      -----------------------------------------------------------------------
      94905259844757031250000000d27 |

              1       1
o16 : Matrix S  <--- S

See also

Ways to use installHilbertFunction :