# How to write Buchberger algorithm?

Hello, I want to write this algorithm in sage, I know sage is python base but I'm not much familiar with this programming language so I'm working on it.. Could you please tell me how can I write Buchberger algorithm in sage? I know there is commands for computing it, but I want the algorithm..

input= (f1,,fs)

output= a groebner basis G={ g1,...,gt} for f ? G

initialization :

G=F

g:= {(fi,fj) | fi,fj ? G , fi!= fj }

h:=0

iteration

    WHILE *g* != 0   DO

choose any {f,g} ?  *g*

*g*:= *g* \ {{f,g}}

h:= (s(f,g)) ^G

IF h != 0  THEN

*g* := *g* U {{u,h}| u ? G }

G:= G U {h}

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A simpler way to find where buchberger() function is located could be the use of the import_statements() function:

sage: import_statements('buchberger')
from sage.rings.polynomial.toy_buchberger import buchberger
sage: from sage.rings.polynomial.toy_buchberger import buchberger
sage: buchberger??

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Check if it is in Sage's source code:

search_doc('buchberger')


This reveals that there is a file

$SAGE_ROOT/devel/sage/sage/rings/polynomial/toy_buchberger.py  (where$SAGE_ROOT is your Sage root directory).

In this file there is a function buchberger and a function buchberger_improved.

To get the code for Buchberger's algorithm, type

sage: from sage.rings.polynomial.toy_buchberger import buchberger
sage: buchberger??


It uses a function to compute the S-polynomial of two polynomials, whose source code you can see by typing

sage: from sage.rings.polynomial.toy_buchberger import spol
sage: spol??


You can also browse the source code online at github. See toy_buchberger there.

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