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installHilbertFunction -- install a Hilbert function without computation

Synopsis

Usage:
installHilbertFunction(M,hf)
Inputs:
- M, an ideal, a module, or a matrix
- hf
Consequences:
- The poincare polynomial hf is used as the poincare polynomial for M

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 0x8cda5f0

   -- [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 0.032002 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 0x8cda000

   -- [gb]number of (nonminimal) gb elements = 0
   -- number of monomials                = 0
   -- 
o9 : Ideal of R

i10 : time hf = poincare I

   -- registering gb 5 at 0x8b75d10

   -- [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 0.020001 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 0x8b755f0

   -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)m
removing gb 1 at 0x8cda4c0
mm{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
   --      -- used 0.492031 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

selectInSubring -- select columns in a subring
Elimination -- eliminating specified variables, and Sylvester resultant

Ways to use `installHilbertFunction` :

installHilbertFunction(Ideal,RingElement)
installHilbertFunction(Matrix,RingElement)
installHilbertFunction(Module,RingElement)