### A.2.2 Groebner basis conversion

The performance of Buchberger's algorithm is sensitive to the chosen monomial order. A Groebner basis computation with respect to a less favorable order such as the lexicographic ordering may easily run out of time or memory even in cases where a Groebner basis computation with respect to a more efficient order such as the degree reverse lexicographic ordering is very well feasible. Groebner basis conversion algorithms and the Hilbert-driven Buchberger algorithm are based on this observation:

- Groebner basis conversion: Given an ideal and a slow monomial order, compute a Groebner basis with respect to an appropriately chosen fast order. Then convert the result to a Groebner basis with respect to the given slow order.

- Hilbert-driven Buchberger algorithm: Homogenize the given generators for with respect to a new variable, say, . Extend the given slow ordering on to a global product ordering on . Compute a Groebner basis for the ideal generated by the homogenized polynomials with respect to a fast ordering. Read the Hilbert function, and use this information when computing a Groebner basis with respect to the extended (slow) ordering. Finally, dehomogenize the elements of the resulting Groebner basis.

SINGULAR provides implementations for the FGLM conversion algorithm (which applies to zero-dimensional ideals only, see stdfglm) and variants of the Groebner walk conversion algorithm (which works for arbitrary ideals, See frwalk, grwalk_lib). An implementation of the Hilbert-driven Buchberger algorithm is accessible via the `stdhilb` command (see also std).

For the ideal below, `stdfglm` is more than 100 times and `stdhilb` about 10 times faster than `std`.

      ring r =32003,(a,b,c,d,e),lp;
      ideal i=a+b+c+d, ab+bc+cd+ae+de, abc+bcd+abe+ade+cde,
              abc+abce+abde+acde+bcde, abcde-1;
      int t=timer;
      option(prot);
      ideal j1=stdfglm(i);
    ==> std in (32003),(a,b,c,d,e),(dp(5),C)
    ==> [4095:2]1(4)s2(3)s3(2)s4s(3)s5s(4)s(5)s(6)6-ss(7)s(9)s(11)-7-ss(13)s(15)s\
       (17)--s--8-s(16)s(18)s(20)s(23)s(26)-s(23)-------9--s(16)s10(19)s(22)s(25\
       )----s(24)--s11---------s12(17)s(19)s(21)------s(17)s(19)s(21)s13(23)s--s\
       -----s(20)----------14-s(12)--------15-s(6)--16-s(5)--17---
    ==> (S:21)---------------------
    ==> product criterion:109 chain criterion:322
    ==> .....+....-...-..-+-....-...-..---...-++---++---....-...-++---.++-----------...------....-...------+--------+---.++------++++-+++----------------+---
    ==> vdim= 45
    ==> .............................................++--------------------------------------------+--------------------------------------------+--------------------------------------------+--------------------------------------------
      timer-t;
    ==> 0
      size(j1);   // size (no. of polys) in computed GB
    ==> 5
      t=timer;
      ideal j2=stdhilb(i);
    ==> compute hilbert series with std in ring (32003),(a,b,c,d,e,@),(dp(6),C)
    ==> weights used for hilbert series: 1,1,1,1,1,1
    ==> [1023:1]1(4)s2(3)s3(2)s4ss5(3)s(4)s(5)-s6s(6)s(7)s(9)s(11)-7-ss(13)s(15)s\
       (17)--s--8-s(16)s(18)s(20)s(23)s(26)-s(29)-------9-s(25)s(28)--s(29)---s-\
       ------10-s(24)-------s(19)---11-s(17)s(19)s(21)-----s(18)-s(19)s12(21)s(2\
       3)s(26)-s(27)------s(23)----------13-s(15)-----------14-s(6)--15-s(5)--16\
       ---
    ==> product criterion:88 chain criterion:650
    ==> std with hilb in (32003),(a,b,c,d,e,@),(lp(5),dp(1),C)
    ==> [1023:1]1(41)s2(40)s3(39)s4s(40)-s5(41)s(44)s(46)s-s-sh6s(49)s(51)s(54)s(\
       55)s(56)s(58)s(59)--shhhhhhh7(53)s(55)s(57)s(59)s(61)-s(62)s(68)s(70)s(71\
       )s(74)--shhhhhhhhhhhhhhhh8(58)s(61)s(65)s(68)s(71)-s(72)s(75)--------shhh\
       hhhhhhhhhhhhhhhh9(51)s(53)s(56)s(58)s(61)s(64)------s(61)s(64)shhhhhhhhhh\
       hhhhh10(53)s(55)s(58)s(62)s(64)s(67)s(70)--s(71)------s(68)s(71)s(73)--sh\
       hhhhhhhhhhhhhhh11(58)s(60)s(63)s(66)s(69)s(72)s(74)---s-s(76)s(79)----s(7\
       8)-------shhhhhhhhhhhhhhhhhhhhhhh12(51)s(54)s(57)s(58)s(60)s(63)s(65)s(68\
       )s(70)s(73)s(76)s(79)--s(80)----shhhhhhhhhhhhhhhhhhhhhhhhhhhhhh13(48)s(51\
       )s(54)s(57)s(59)s(61)s(64)s(67)shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh14\
       (31)s(33)s(36)s(39)s(42)s(45)shhhhhhhhhhhhhhhhhhhhhhhhh15(23)s(26)s(29)s(\
       32)s(35)shhhhhhhhhhhhhhhhhhh16(18)s(21)s(24)s(27)shhhhhhhhhhhhhhh17(15)s(\
       18)s(21)s(24)shhhhhhhhhhhh18(15)s(18)s(21)s(24)shhhhhhhhhhhh19(14)s(17)s(\
       20)shhhhhhhhhhhh20(11)s(14)s(17)shhhhhhhhh21(11)s(14)s(17)shhhhhhhhh22(11\
       )s(14)s(16)shhhhhhhhh23(10)s(13)shhhhhhhhh24(7)s(10)shhhhhh25(7)s(10)shhh\
       hhh26(7)s(10)shhhhhh27(7)s(10)shhhhhh28(7)s(10)shhhhhh29(7)s(10)shhhhhh30\
       (7)s(9)shhhhhh31(6)shhhhhh32(3)shhh33shhh34shhh35shhh36shhh37shhh38shhh39\
       shhh40shhh41shhh42shhh43shhh44shhh45shhh46shhh47shhh48shhh49shhh50shhh51s\
       hhh52shhh53shhh54shhhhhh
    ==> product criterion:491 chain criterion:11799
    ==> hilbert series criterion:417
    ==> dehomogenization
    ==> simplification
    ==> imap to ring (32003),(a,b,c,d,e),(lp(5),C)
      timer-t;
    ==> 0
      size(j2);   // size (no. of polys) in computed GB
    ==> 5
      // usual Groebner basis computation for lex ordering
      t=timer;
      ideal j0 =std(i);
    ==> [4095:1]1(4)s2(3)s3(2)s4s(3)s5(5)s(4)s6(6)s(7)s(9)s(8)sss7(10)s(11)s(10)s\
       (11)s(13)s8(12)s(13)s(15)s.s(14).s.9.s(16)s(17)s(19)........10.s(20).s(21\
       )ss..11.s(23)s(25).ss(27)...s(28)s(26)...12.s(25)sss(23)sss.......s(22)..\
       .13.s(23)ssssssss(21)s(22)sssss(21)ss..14.ss(22)s.s.sssss(21)s(22)sss.s..\
       .15.ssss(21)s(22)ssssssssss(21)s(22)sss16.ssssssss(21)s(22)sssssssssss17s\
       s(21)s(22)ssssssssss(21)sss(22)ss(21)ss18(22)s(21)s(22)s.s..............1\
       9.sssss(21)ss(22)ssssssssss(21)s(22)s20.ssssssssss(21)s........21.s(22)ss\
       sssssssssssss(21)s(22)ssss22ssssssssssss(21)s(22)sssssss23sssssssssss(21)\
       s(22)ssssssss24ssssssssssss(21)s(22)sssssss25ssssssssss(21)s(22)sssssssss\
       26ssssssssss(21)s(20)ssssssss27.sssssssss..........s28.ssssss............\
       .29.sssssssssssssssssss30sssssssssssssssssss31.sssssssssssssssssss32.ssss\
       ssssssssssssssss33ssssssssssssssssssss34ssssssssssssssssssss35sssssssssss\
       sssssssss36ssssssssssssssssssss37ssssssssssssssssssss38ssssssssssssssssss\
       ss39ssssssssssssssssssss40ssssssssssssssssssss41ssss---------------42-s(4\
       )--43-s44s45s46s47s48s49s50s51s52s53s54s55s56s
    ==> product criterion:1395 chain criterion:904
      option(noprot);
      timer-t;
    ==> 0

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