# Algorithm for Grobner Basis

We can calculate Groebner Basis as follows: I want to know the corresponding algorithm. Is it F5?

sage :  P.<x ,y> = PolynomialRing (GF(11) ,order =’lex ’)

sage : f1 = (x -1)^2 + (y -2)^2 - 3^2

sage : f2 = ( x +1)^2 + (y -1)^2 - 2^2

sage : I = Ideal ( f1 , f2 )

sage : I. groebner_basis ()

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Specific algorithm can be requested via .groebner_basis(algorithm=...).

( 2022-04-10 17:42:15 +0200 )edit

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In your case, we can say that Singular library is called, see:

sage: from sage.misc.citation import get_systems
sage: get_systems('I.groebner_basis()')
['Singular']


But then, i am not sure which algorithm is called there, though apparently Singular ships F5, see https://en.wikipedia.org/wiki/Faugère...

Note that you can chose which implementation, see

sage: I.groebner_basis?


For example, you can specify (i just pick a random algorithm among all possible choices):

sage: I.groebner_basis(algorithm='libsingular:slimgb')
[x - 5*y - 5, y^2 - 2*y]


Though the name slimgbis not particularly self-contained (see https://singular.mathematik.uni-kl.de... more details).

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