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Grobner bases of ideals

asked 2016-12-21 06:11:33 -0500

Nilesh gravatar image

updated 2016-12-22 06:31:42 -0500

tmonteil gravatar image

Respected Sir, I am trying to find Groebner basis of an ideal in polynomial ring in 35 variables over GF(2)(As per suggestions earlier, I am working over GF(2) instead of GF(3), since in GF(2) coefficient growth is restricted) but I am not able to see the output using sage. Even it do not shows any error in it. So,how to get the output?(Even I tried singular, but can't succeed.) .Even I tried using degrevlex, but can not get any output. Following is the code w.r.t. lex ordering:

P.<x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35>=PolynomialRing(FiniteField(2),order='lex');        
I=Ideal([x1*x21*x22*x23*x24*x25*x26*x27*x28*x29*x30*x31*x32*x33*x34*x35-1,x2*x14*x15*x16*x17*x18*x19*x20*x21*x22*x23*x24*x25*x26*x27*x28-1,x3*x11*x12*x13*x15*x17*x19*x20*x21*x22*x23*x24*x29*x30*x31*x35-1,x4*x8*x9*x10*x14*x18*x19*x20*x23*x24*x25*x27*x29*x30*x32*x34-1,x5*x7*x8*x10*x12*x13*x19*x20*x22*x24*x25*x26*x29*x31*x32*x33-1,x6*x7*x9*x10*x11*x13*x16*x17*x18*x20*x22*x23*x25*x26*x29*x30-1,x1^2-1,x2^2-1,x3^2-1,x4^2-1,x5^2-1,x6^2-1,x7^2-1,x8^2-1,x9^2-1,x10^2-1,x11^2-1,x12^2-1,x13^2-1,x14^2-1,x15^2-1,x16^2-1,x17^2-1,x18^2-1,x19^2-1,x20^2-1,x21^2-1,x22^2-1,x23^2-1,x24^2-1,x25^2-1,x26^2-1,x27^2-1,x28^2-1,x29^2-1,x30^2-1,x31^2-1,x32^2-1,x33^2-1,x34^2-1,x35^2-1]);    

I.groebner_basis();
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Could you please provide all the commands you typed. How did you create your polynomial ring, how did you create your ideal, etc. We can not help debugging otherwise.

tmonteil gravatar imagetmonteil ( 2016-12-21 06:43:50 -0500 )edit

Dear Sir, as per your suggestion, I edited my earlier question and given the entire code, which I used.

Nilesh gravatar imageNilesh ( 2016-12-21 22:40:09 -0500 )edit

Note: also asked on sage-support.

slelievre gravatar imageslelievre ( 2016-12-22 07:10:05 -0500 )edit

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answered 2017-01-13 02:25:19 -0500

asante gravatar image

When you have unknowns in GF(2), you might want to use BooleanPolynomialRing, instead of the more general PolynomialRing. The BooleanPolynomialRing takes care of that for all unknowns in GF(2) it holds: x^2 = x. With this you code looks like this:

P.<x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35>=BooleanPolynomialRing(35,order='lex');
I=Ideal([x1*x21*x22*x23*x24*x25*x26*x27*x28*x29*x30*x31*x32*x33*x34*x35-1,x2*x14*x15*x16*x17*x18*x19*x20*x2*x22*x23*x24*x25*x26*x27*x28-1,x3*x11*x12*x13*x15*x17*x19*x20*x21*x22*x23*x24*x29*x30*x31*x35-1,x4*x8*x9*x10*x14*x18*x19*x20*x23*x24*x25*x27*x29*x30*x32*x34-1,x5*x7*x8*x10*x12*x13*x19*x20*x22*x24*x25*x26*x29*x31*x32*x33-1,x6*x7*x9*x10*x11*x13*x16*x17*x18*x20*x22*x23*x25*x26*x29*x30-1,x1^2-1,x2^2-1,x3^2-1,x4^2-1,x5^2-1,x6^2-1,x7^2-1,x8^2-1,x9^2-1,x10^2-1,x11^2-1,x12^2-1,x13^2-1,x14^2-1,x15^2-1,x16^2-1,x17^2-1,x18^2-1,x19^2-1,x20^2-1,x21^2-1,x22^2-1,x23^2-1,x24^2-1,x25^2-1,x26^2-1,x27^2-1,x28^2-1,x29^2-1,x30^2-1,x31^2-1,x32^2-1,x33^2-1, x34^2-1,x35^2-1])
I.groebner_basis();

My laptop computes this Groebner basis within seconds.

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Asked: 2016-12-21 06:11:33 -0500

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Last updated: Dec 22 '16