### 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 (ZZ/32003),(a,b,c,d,e),(dp(5),C) ==> [1048575:3]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(1\ 5)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 (ZZ/32003),(a,b,c,d,e,@),(dp(6),C\ ) ==> weights used for hilbert series: 1,1,1,1,1,1 ==> [1048575:2]1(4)s2(3)s3(2)s4ss5(3)s(4)s(5)-s6s(6)s(7)s(9)s(11)-7-ss(13)s(1\ 5)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(23)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 (ZZ/32003),(a,b,c,d,e,@),(lp(5),dp(1),C) ==> [1048575:2]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)--------s\ hhhhhhhhhhhhhhhhhhh9(51)s(53)s(56)s(58)s(61)s(64)------s(61)s(64)shhhhhhh\ hhhhhhhh10(53)s(55)s(58)s(62)s(64)s(67)s(70)--s(71)------s(68)s(71)s(73)-\ -shhhhhhhhhhhhhhhh11(58)s(60)s(63)s(66)s(69)s(72)s(74)---s-s(76)s(79)----\ s(78)-------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)shhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh\ h14(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)s\ hhhhhh26(7)s(10)shhhhhh27(7)s(10)shhhhhh28(7)s(10)shhhhhh29(7)s(10)shhhhh\ h30(7)s(9)shhhhhh31(6)shhhhhh32(3)shhh33shhh34shhh35shhh36shhh37shhh38shh\ h39shhh40shhh41shhh42shhh43shhh44shhh45shhh46shhh47shhh48shhh49shhh50shhh\ 51shhh52shhh53shhh54shhhhhh ==> product criterion:491 chain criterion:11799 ==> hilbert series criterion:417 ==> dehomogenization ==> simplification ==> imap to ring (ZZ/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); ==> [1048575:2]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(1\ 0)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)sssssssssss\ 17ss(21)s(22)ssssssssss(21)sss(22)ss(21)ss18(22)s(21)s(22)s.s............\ ..19.sssss(21)ss(22)ssssssssss(21)s(22)s20.ssssssssss(21)s........21.s(22\ )sssssssssssssss(21)s(22)ssss22ssssssssssss(21)s(22)sssssss23sssssssssss(\ 21)s(22)ssssssss24ssssssssssss(21)s(22)sssssss25ssssssssss(21)s(22)ssssss\ sss26ssssssssss(21)s(20)ssssssss27.sssssssss..........s28.ssssss.........\ ....29.sssssssssssssssssss30sssssssssssssssssss31.sssssssssssssssssss32.s\ sssssssssssssssssss33ssssssssssssssssssss34ssssssssssssssssssss35ssssssss\ ssssssssssss36ssssssssssssssssssss37ssssssssssssssssssss38sssssssssssssss\ sssss39ssssssssssssssssssss40ssssssssssssssssssss41ssss---------------42-\ s(4)--43-s44s45s46s47s48s49s50s51s52s53s54s55s56s ==> product criterion:1395 chain criterion:904 option(noprot); timer-t; ==> 0 ```

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