With no Strategy option, this just calls gb.
i1 : R = QQ[a..d] o1 = R o1 : PolynomialRing |
i2 : M = groebnerBasis random(R^1,R^{4:-2});
1 12
o2 : Matrix R <--- R
|
i3 : netList (ideal M)_*
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 |
o3 = |11299050a*c + 6906531b*c + 5601225c + 22096970a*d + 10053225b*d - 10138695c*d - 803376d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|9039240b + 154881b*c + 789975c + 2230550a*d + 2703015b*d + 532455c*d + 1175328d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 |
|45196200a*b - 2066697b*c - 3338775c - 12617990a*d - 9788625b*d + 32683665c*d + 1493072d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|45196200a - 6222819b*c - 4176525c - 40248530a*d - 6956775b*d + 33630555c*d + 7946624d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 |
|3964358137052399637161262316798260c d + 594168029286400198536644282613264a*d + 1337598604362667816823185742844123b*d + 4247534217639439790300643591761688c*d + 633197914362837742374002422591576d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 2 3 |
|792871627410479927432252463359652b*c*d + 907626632807183390740981849104360a*d + 459506552743851049077630547055715b*d - 482254086694892198398595134531980c*d - 66868686610007296427523927539912d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 3 2 2 2 3|
|95144595289257591291870295603158240c - 2841837152455056818005034904179920616a*d - 976817993612564469525108588611526519b*d + 2581638507104454154462789583156791212c*d + 363076509596312359223810846059182696d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 |
|19028919057851518258374059120631648b*c + 64257621219904726957716712697169784a*d + 33694597027785012215432150401502373b*d - 62238924479127557348547050385679908c*d - 7226409689483732200678023735720504d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 3 4 |
|92377306833998610621043459166544878258614194423854521609030515753c*d + 8498151665540849564229808863382997782054706833761464637212817128d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 3 4 |
|461886534169993053105217295832724391293070972119272608045152578765b*d + 90396066568414181641497131332966112726627833129769050446288485072d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 3 4 |
|153962178056664351035072431944241463764356990706424202681717526255a*d - 17326088189814257165217320650455953486068430536610639471432672813d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| 5 |
|d |
+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|
With a Strategy option, the code is experimental, subject to interface changes, and might have bugs. So use at your own risk! However, it appears to work correctly and is often very fast, in cases where it applies. If you encounter any bugs, please let us know!
If either "MGB" (MGB stands for mathicGB, the name of the package used), or "F4" is given for the Strategy, then experimental code (written by Bjarke Roune and M. Stillman) is used. The plan is for this to become the default version for Groebner bases in later versions of Macaulay2. But for now, it is experimental.
These strategies only work for ideals in polynomial rings over a finite field ZZ/p. In other cases, either an error will be given, or the current default Groebner basis algorithm will be used.
i4 : R = ZZ/101[a..e] o4 = R o4 : PolynomialRing |
i5 : I = ideal sub(random(R^1, R^{4:-2}), e=>1);
o5 : Ideal of R
|
i6 : netList I_*
+------------------------------------------------------------------------------------------------------+
| 2 2 2 2 |
o6 = |- 33a - 19a*b - 39b + 17a*c + 36b*c + 4c - 20a*d + 9b*d + 13c*d + 22d + 44a - 39b - 26c - 49d - 11|
+------------------------------------------------------------------------------------------------------+
| 2 2 2 2 |
|- 8a + 43a*b - 22b - 8a*c - 30b*c - 28c + 36a*d + 41b*d - 6c*d - 9d - 3a + 16b + 35c - 35d + 6 |
+------------------------------------------------------------------------------------------------------+
| 2 2 2 2 |
|40a + 3a*b - 41b - 31a*c - 49b*c + 30c + 25a*d - 13b*d - 47c*d - 40d - 2a + 4b + 27c + 37d - 35 |
+------------------------------------------------------------------------------------------------------+
| 2 2 2 2 |
|- 31a - 39a*b - 48b - 31a*c + 30b*c - 49c - 48a*d - 37b*d + 28c*d + 46d - 29a + 47b - 18c + d + 40|
+------------------------------------------------------------------------------------------------------+
|
i7 : gbI = ideal groebnerBasis(I, Strategy=>"MGB"); o7 : Ideal of R |
i8 : netList gbI_*
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 |
o8 = |a*c + 12b*c - 46c + 43a*d + 33b*d - 26c*d - 3d - 15a + 42b + 49c - 13 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|b + 28b*c + 40c + 28a*d - 11b*d + 35c*d - 13d - 29a + 18b - 15c - 17d + 15 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 |
|a*b + 21b*c + 15c + 26a*d + 42b*d + 46c*d - 34d - 32a + 8b + 38c + 14d - 49 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|a + 15b*c - 43c - 10a*d - 22b*d + c*d - 4d - 39a + 28c + 38d - 2 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 2 2 |
|c d - 34a*d + 37b*d + 29c*d + 42d + 10b*c - 11c + 17a*d + 9b*d + 32c*d + 8d - 39a - 36b + 32c + 25d - 49 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 3 2 2 |
|b*c*d - 22a*d + 5b*d + 42c*d - 21d - 43b*c - 36c - 2a*d - 13b*d - 3c*d + 25d + 7a + 11b - 37c + 40d - 22 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 2 2 2 3 2 2 |
|c - 31a*d + 30b*d - 22c*d - 29d + 12b*c + 34c + 41a*d - b*d - 27c*d + 33d - 13a - 21b - 49c - 29d - 24 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 2 2 |
|b*c + 19a*d + 2b*d - 16c*d - d - 35b*c + 32c - 19a*d - 33b*d - 24c*d - 37d + 47a - 33b - 31c - 28d - 12 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 3 2 2 |
|c*d - 43d - 33a*d - 12b*d + 7d - 18b*c - 40c - 16a*d - 5b*d - 5c*d + 30d + 32a - 26b - 43c + 20d + 34 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 2 3 2 2 |
|b*d - 32d - 16a*d + 3b*d - 34c*d - 33d + b*c + 24c + 39a*d - b*d - 45c*d + 13d - 49a + 18b - 3c + 2d + 34 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 2 3 2 2 |
|a*d - 15d + 10a*d - 25b*d - 43c*d + 21d - 15b*c + 46c - 3a*d - b*d - 5c*d - 8d - 29a + 19b + 30c - 8d + 21 |
+-------------------------------------------------------------------------------------------------------------------+
| 5 4 2 2 2 3 2 2 |
|d + 34d + 41a*d - 21b*d - 39c*d + 10d - 16b*c - 44c - 13a*d + 47b*d + c*d + 44d - 29a - 18b + 25c + 2d - 19|
+-------------------------------------------------------------------------------------------------------------------+
|
Also implemented is a Faugere-like algorithm that is sometimes much faster (but also sometimes takes a large amount of memory).
i9 : gbTrace=1 o9 = 1 |
i10 : gbI = ideal groebnerBasis(I, Strategy=>"F4");
-- computing mgb F4 OptionTable{Log => }
Reducer => F4
SPairGroupSize => 0
Threads => 0
o10 : Ideal of R
|
i11 : netList gbI_*
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 |
o11 = |a*c + 12b*c - 46c + 43a*d + 33b*d - 26c*d - 3d - 15a + 42b + 49c - 13 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|b + 28b*c + 40c + 28a*d - 11b*d + 35c*d - 13d - 29a + 18b - 15c - 17d + 15 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 |
|a*b + 21b*c + 15c + 26a*d + 42b*d + 46c*d - 34d - 32a + 8b + 38c + 14d - 49 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 |
|a + 15b*c - 43c - 10a*d - 22b*d + c*d - 4d - 39a + 28c + 38d - 2 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 2 2 |
|c d - 34a*d + 37b*d + 29c*d + 42d + 10b*c - 11c + 17a*d + 9b*d + 32c*d + 8d - 39a - 36b + 32c + 25d - 49 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 3 2 2 |
|b*c*d - 22a*d + 5b*d + 42c*d - 21d - 43b*c - 36c - 2a*d - 13b*d - 3c*d + 25d + 7a + 11b - 37c + 40d - 22 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 2 2 2 3 2 2 |
|c - 31a*d + 30b*d - 22c*d - 29d + 12b*c + 34c + 41a*d - b*d - 27c*d + 33d - 13a - 21b - 49c - 29d - 24 |
+-------------------------------------------------------------------------------------------------------------------+
| 2 2 2 2 3 2 2 |
|b*c + 19a*d + 2b*d - 16c*d - d - 35b*c + 32c - 19a*d - 33b*d - 24c*d - 37d + 47a - 33b - 31c - 28d - 12 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 3 2 2 |
|c*d - 43d - 33a*d - 12b*d + 7d - 18b*c - 40c - 16a*d - 5b*d - 5c*d + 30d + 32a - 26b - 43c + 20d + 34 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 2 3 2 2 |
|b*d - 32d - 16a*d + 3b*d - 34c*d - 33d + b*c + 24c + 39a*d - b*d - 45c*d + 13d - 49a + 18b - 3c + 2d + 34 |
+-------------------------------------------------------------------------------------------------------------------+
| 3 4 2 2 2 3 2 2 |
|a*d - 15d + 10a*d - 25b*d - 43c*d + 21d - 15b*c + 46c - 3a*d - b*d - 5c*d - 8d - 29a + 19b + 30c - 8d + 21 |
+-------------------------------------------------------------------------------------------------------------------+
| 5 4 2 2 2 3 2 2 |
|d + 34d + 41a*d - 21b*d - 39c*d + 10d - 16b*c - 44c - 13a*d + 47b*d + c*d + 44d - 29a - 18b + 25c + 2d - 19|
+-------------------------------------------------------------------------------------------------------------------+
|
(1) The MGB and F4 options are experimental, work only over a finite field of char $< 2^{32}$, not over quotient rings, and not over exterior or Weyl algebras. However, these versions can be much faster when they apply. (2) The experimental versions do not stash their results into the ideal or module. (3) The experimental version only works for ideals currently.
The object groebnerBasis is a method function with options.