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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 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 0x93a2260

   -- [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                = 4182
   -- ncalls = 10
   -- nloop = 29
   -- nsaved = 0
   --      -- used 0.03 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 Gröbner basis using a different monomial order. In the example below

i8 : R = QQ[a..d];

   -- registering polynomial ring 5 at 0x93d3510
i9 : I = ideal random(R^1, R^{3:-3});

   -- registering gb 4 at 0x8eabab0

   -- [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 0x8eab980

   -- [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 0x93d3480

o11 = S

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

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

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

o14 = 3
i15 : time gens gb J;

   -- registering gb 6 at 0x8eab130

   -- [gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)
removing gb 1 at 0x93a2130
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 0.82 seconds

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

o16 = | 468980212538660459821847284479473732469283738198048554020961517544699
      -----------------------------------------------------------------------
      6308060800000000c27+486545994075471725864855590677318286099796432779663
      -----------------------------------------------------------------------
      83782797650520686764685825600000000c26d+
      -----------------------------------------------------------------------
      22720960729801956463179408112397883830706568969169943767422345714567392
      -----------------------------------------------------------------------
      1372314384000000c25d2+6271726410173581654551678011357787412251889645756
      -----------------------------------------------------------------------
      10607222571752438223367923192376400000c24d3+
      -----------------------------------------------------------------------
      19875236893097373453006898256148280007684260327042269087351968047402999
      -----------------------------------------------------------------------
      07524958059500000c23d4+
      -----------------------------------------------------------------------
      24401502557445671077292532546752110367169311059935187293781493987305169
      -----------------------------------------------------------------------
      29479559025760000c22d5-
      -----------------------------------------------------------------------
      94167194750828694475642151429331746234309172001713522612414087529528674
      -----------------------------------------------------------------------
      39588772698804375c21d6-
      -----------------------------------------------------------------------
      34901005194115290185971717973576234610066445123246075169777431015699870
      -----------------------------------------------------------------------
      238080951070631000c20d7-
      -----------------------------------------------------------------------
      35192454082341079668358214931291442935950502749908035906542222741265258
      -----------------------------------------------------------------------
      845642212082774250c19d8+
      -----------------------------------------------------------------------
      21508078672624917984595616105262807867028517595351376768646399636110499
      -----------------------------------------------------------------------
      475308891320082600c18d9+
      -----------------------------------------------------------------------
      89198294016160748858176088802731608612365134763631376484520038296435933
      -----------------------------------------------------------------------
      833906939923316300c17d10+
      -----------------------------------------------------------------------
      96193886435707238441608683569969036909978629086846599647089983333635486
      -----------------------------------------------------------------------
      901440659741302640c16d11+
      -----------------------------------------------------------------------
      32375808939012230007051298710480506617583615967324441807806820722955782
      -----------------------------------------------------------------------
      935995984712657610c15d12-
      -----------------------------------------------------------------------
      62015151178897855972698604964496564971277649125041821256807306825352598
      -----------------------------------------------------------------------
      886396086449466225c14d13-
      -----------------------------------------------------------------------
      12885464671715404778033832248526294704170557027332502092132143714777542
      -----------------------------------------------------------------------
      5553967466526180432c13d14-
      -----------------------------------------------------------------------
      12642505225139437321059158967537499654797869990027299952369709209089576
      -----------------------------------------------------------------------
      1381476099534471912c12d15-
      -----------------------------------------------------------------------
      75168769604614373061727841167426267954694802343093379471949771277182341
      -----------------------------------------------------------------------
      099189219710215600c11d16-
      -----------------------------------------------------------------------
      21598951646502372886051569171490875529619939423848982193224270058920283
      -----------------------------------------------------------------------
      892687510744449696c10d17+
      -----------------------------------------------------------------------
      14284043424710255841975190831246317775759290642889261364124255085060291
      -----------------------------------------------------------------------
      140902378396852880c9d18+
      -----------------------------------------------------------------------
      26976976767979572334252386542199328554997330149073369330071211729273785
      -----------------------------------------------------------------------
      225721583078313696c8d19+
      -----------------------------------------------------------------------
      21204600518460887322078439147419499201024805794853214548347572466958034
      -----------------------------------------------------------------------
      309010152422263584c7d20+
      -----------------------------------------------------------------------
      11957161105805242856102967191538841229383858794039985856957856336785238
      -----------------------------------------------------------------------
      433755112666303488c6d21+
      -----------------------------------------------------------------------
      60847808374957577651036905814934121957812163763784400949176588579813480
      -----------------------------------------------------------------------
      83972315209104896c5d22+
      -----------------------------------------------------------------------
      28241685938428907339258508857758569305429279265776192319862990121890934
      -----------------------------------------------------------------------
      51043277826566848c4d23+
      -----------------------------------------------------------------------
      16468111878793480066452263800050597436537514341898268480991478319165208
      -----------------------------------------------------------------------
      18582619590571520c3d24+
      -----------------------------------------------------------------------
      11533890162545562835113842353065179067446032531767096411131575622409882
      -----------------------------------------------------------------------
      96956089889865600c2d25+
      -----------------------------------------------------------------------
      48131905844014750090072417357186544597870179487719572511695794005616208
      -----------------------------------------------------------------------
      6382953819216000cd26+72206164434669479842272170683151133922642535840284
      -----------------------------------------------------------------------
      856874208762970113274309794289688000d27 |

              1       1
o16 : Matrix S  <--- S

See also

Ways to use installHilbertFunction :