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Buchberger Algorithm

asked 2013-02-17 06:23:46 -0500

Neda gravatar image

updated 2013-02-17 09:05:34 -0500

calc314 gravatar image

Hi! could you please tell me which command I should use for contributing the buchberger algorithm to finding a groebner basis for an Ideal over a field like rational field? I found these commands but did'nt work..

sage: from sage.rings.polynomial.toy_buchberger import *
sage: P.<a,b,c,e,f,g,h,i,j,k> = PolynomialRing(GF(32003),10)
sage: I = sage.rings.ideal.Katsura(P,6)

sage: g1 = buchberger(I)
sage: g2 = buchberger_improved(I)
sage: g3 = I.groebner_basis()
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answered 2013-02-18 09:15:12 -0500

Mike Hansen gravatar image

The best option is to just use I.groebner_basis().

sage: P.<a,b,c,e,f,g,h,i,j,k> = PolynomialRing(GF(32003),10)
sage: I = sage.rings.ideal.Katsura(P,6)
sage: I.groebner_basis()
Polynomial Sequence with 22 Polynomials in 6 Variables
sage: list(_)
[g^6 - 11466*g^5 - 3669*b*g^3 + 4465*c*g^3 - 2315*e*g^3 + 5098*f*g^3 - 8372*g^4 - 15837*f^3 - 9547*b*e*g + 839*b*f*g + 15210*c*f*g - 2141*f^2*g - 13725*b*g^2 - 10720*c*g^2 - 8122*e*g^2 - 2102*f*g^2 - 13908*g^3 + 4663*b*e + 12265*b*f + 6285*c*f + 12921*e*f + 11846*f^2 + 6867*b*g + 9878*c*g - 860*e*g - 1743*f*g - 8368*g^2 + 1892*b + 8121*c - 3112*e + 4349*f + 2284*g, b*g^4 + 12019*g^5 + 6465*b*g^3 - 13833*c*g^3 - 14150*e*g^3 - 1091*f*g^3 - 14524*g^4 - 15914*f^3 - 15812*b*e*g + 15278*b*f*g - 4542*c*f*g - 7300*f^2*g + 9977*b*g^2 - 14014*c*g^2 + 3296*e*g^2 + 5812*f*g^2 - 10065*g^3 - 4292*b*e - 463*b*f - 12423*c*f - 853*e*f - 11889*f^2 + 3839*b*g + 6212*c*g - 7338*e*g + 1142*f*g + 12073*g^2 + 15253*b - 7437*c + 11862*e - 6614*f + 11312*g, c*g^4 - 13467*g^5 + 14690*b*g^3 - 1071*c*g^3 + 7136*e*g^3 + 249*f*g^3 + 5472*g^4 + 11550*f^3 + 3557*b*e*g + 1925*b*f*g + 9868*c*f*g - 130*f^2*g - 3432*b*g^2 - 6768*c*g^2 + 1982*e*g^2 + 12826*f*g^2 - 12180*g^3 + 13808*b*e + 11322*b*f + 2171*c*f + 12372*e*f + 10928*f^2 + 750*b*g + 9052*c*g - 9008*e*g - 2003*f*g - 5879*g^2 - 15186*b + 8220*c - 6034*e - 7337*f + 906*g, e*g^4 - 13178*g^5 - 2372*b*g^3 + 2247*c*g^3 - 14789*e*g^3 + 2351*f*g^3 - 6361*g^4 + 8004*f^3 - 3036*b*e*g + 11459*b*f*g - 15592*c*f*g - 1543*f^2*g - 903*b*g^2 - 9248*c*g^2 - 10163*e*g^2 - 10066*f*g^2 - 6929*g^3 + 8587*b*e + 3419*b*f - 12100*c*f - 6787*e*f - 10573*f^2 - 2036*b*g + 8698*c*g + 7884*e*g - 7506*f*g + 2111*g^2 + 14794*b - 15398*c + 13481*e - 3213*f + 2440*g, f*g^4 - 3764*g^5 - 14119*b*g^3 + 2796*c*g^3 + 11676*e*g^3 - 3057*f*g^3 + 12155*g^4 + 3551*f^3 - 15434*b*e*g - 1738*b*f*g - 8112*c*f*g - 15180*f ...
(more)
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I didnt get any error..Thank you so much!

Neda gravatar imageNeda ( 2013-02-19 05:46:46 -0500 )edit
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answered 2013-02-17 09:19:55 -0500

calc314 gravatar image

You can call the grobner package in maxima within Sage. Here is one option:

maxima('load(grobner)')
ans=maxima('poly_buchberger([x^2+y^2,x*y-y^2],[x,y])')
ans

You can see the other options in the grobner package here.

There may be a wrapper for these options already written into Sage, but I haven't found it.

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thank you so much,that was so helpful.

Neda gravatar imageNeda ( 2013-02-19 05:45:11 -0500 )edit

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Asked: 2013-02-17 06:23:46 -0500

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Last updated: Feb 18 '13