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BeniBela
 Post subject: Gröbner basis of polynomials with unknown coefficients
PostPosted: Thu May 19, 2016 6:25 pm 

Joined: Thu May 19, 2016 4:33 pm
Posts: 2
If you have polynomials of certain structure, with unknown coefficients, you can represent the coefficients with invariates, e.g. for a very simple polynomial in x: "x^2 + a x + b" , where we do not know the coefficients a and b.

What is best way to calculate a Gröbner basis of such polynomials in x, while not calculating one in a and b?

More concrete I have equation (polynomial) systems of this structure:

L1*L3*s13-L1*s14-L3*s23+s24
L1*L4*s14-L1*s15-L4*s24+s25
L2*L3*s23-L2*s24-L3*s33+s34
L2*L4*s24-L2*s25-L4*s34+s35
L3*L4*s34-L3*s35-L4*s44+s45

with L variables that are relevant and the s variables as placeholders.

Now I calculate a Gröbner Basis:

Code:
ring r = 0,(L1,L2,L3,L4,L5,L6,L7,L8,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),dp;
groebner(ideal(L1*L3*s13-L1*s14-L3*s23+s24, L1*L4*s14-L1*s15-L4*s24+s25, L2*L3*s23-L2*s24-L3*s33+s34, L2*L4*s24-L2*s25-L4*s34+s35, L3*L4*s34-L3*s35-L4*s44+s45));

_[1]=L3*L4*s34-L3*s35-L4*s44+s45
_[2]=s15*s24*s33-s14*s25*s33-s15*s23*s34+s13*s25*s34+s14*s23*s35-s13*s24*s35
_[3]=L2*L4*s24-L2*s25-L4*s34+s35
_[4]=L2*L3*s23-L2*s24-L3*s33+s34
_[5]=L1*L4*s14-L1*s15-L4*s24+s25
_[6]=L1*L3*s13-L1*s14-L3*s23+s24
_[7]=L2*L4*s23*s44-L2*s25*s34+L2*s24*s35-L4*s33*s44-L2*s23*s45+s33*s45
_[8]=L1*L4*s13*s44-L1*s15*s34+L1*s14*s35-L4*s23*s44-L1*s13*s45+s25*s34-s24*s35+s23*s45
_[9]=L3*s25*s33*s34-L3*s24*s33*s35-L4*s24*s33*s44+L4*s23*s34*s44-s25*s34^2+s24*s34*s35+s24*s33*s45-s23*s34*s45
_[10]=L3*s15*s33*s34-L3*s14*s33*s35-L4*s14*s33*s44+L4*s13*s34*s44-s15*s34^2+s14*s34*s35+s14*s33*s45-s13*s34*s45
_[11]=L2*s24*s25*s34-L2*s24^2*s35-L2*s23*s25*s44+L4*s24*s33*s44-L4*s23*s34*s44+L2*s23*s24*s45+s23*s35*s44-s24*s33*s45
_[12]=L2*L3*s25*s34-L2*L3*s24*s35-L2*s25*s44+L2*s24*s45+s35*s44-s34*s45
_[13]=L3*s15*s23*s34-L3*s13*s25*s34-L3*s14*s23*s35+L3*s13*s24*s35-L4*s14*s23*s44+L4*s13*s24*s44-s15*s24*s34+s14*s25*s34+s14*s23*s45-s13*s24*s45
_[14]=L1*s14*s15*s34-L1*s14^2*s35-L1*s13*s15*s44+L4*s14*s23*s44-L4*s13*s24*s44+L1*s13*s14*s45-s14*s25*s34+s14*s24*s35+s13*s25*s44-s14*s23*s45
_[15]=L1*L3*s15*s34-L1*L3*s14*s35-L3*s25*s34+L3*s24*s35-L1*s15*s44+L1*s14*s45+s25*s44-s24*s45
_[16]=L3*L4*s24*s33-L3*s25*s33-L4*s23*s44+s25*s34-s24*s35+s23*s45
_[17]=L3*L4*s14*s33-L3*s15*s33-L4*s13*s44+s15*s34-s14*s35+s13*s45
_[18]=L1*L2*s15*s24-L1*L2*s14*s25-L1*s15*s34+L1*s14*s35+s25*s34-s24*s35
_[19]=L1*L2*s15*s23-L1*L2*s13*s25-L1*s15*s33+L1*s13*s35+s25*s33-s23*s35
_[20]=L3*L4*s14*s23-L3*L4*s13*s24-L3*s15*s23+L3*s13*s25+s15*s24-s14*s25
_[21]=L1*L2*s14*s23-L1*L2*s13*s24-L1*s14*s33+L1*s13*s34+s24*s33-s23*s34
_[22]=L4^2*s24*s33*s44-L4^2*s23*s34*s44+L4*s25*s34^2-L4*s24*s34*s35-L4*s25*s33*s44+L4*s23*s35*s44-L4*s24*s33*s45+L4*s23*s34*s45-s25*s34*s35+s24*s35^2+s25*s33*s45-s23*s35*s45
_[23]=L4^2*s14*s33*s44-L4^2*s13*s34*s44+L4*s15*s34^2-L4*s14*s34*s35-L4*s15*s33*s44+L4*s13*s35*s44-L4*s14*s33*s45+L4*s13*s34*s45-s15*s34*s35+s14*s35^2+s15*s33*s45-s13*s35*s45
_[24]=L4^2*s14*s23*s44-L4^2*s13*s24*s44+L4*s15*s24*s34-L4*s14*s25*s34-L4*s15*s23*s44+L4*s13*s25*s44-L4*s14*s23*s45+L4*s13*s24*s45-s15*s24*s35+s14*s25*s35+s15*s23*s45-s13*s25*s45
_[25]=L1*L2*s14*s25*s34-L1*L2*s14*s24*s35-L1*L2*s13*s25*s44+L1*L2*s13*s24*s45+L4*s24*s33*s44-L4*s23*s34*s44+L1*s13*s35*s44-L1*s13*s34*s45-s24*s33*s45+s23*s34*s45
_[26]=L2*L4*s14*s25*s33+L2*L4*s15*s23*s34-L2*L4*s13*s25*s34-L2*L4*s14*s23*s35-L2*s15*s25*s33-L4*s15*s33*s34+L2*s13*s25*s35+L4*s13*s34*s35+s15*s33*s35-s13*s35^2
_[27]=L2*s14*s25^2*s33*s34+L2*s15*s23*s25*s34^2-L2*s13*s25^2*s34^2-L2*s14*s24*s25*s33*s35-L2*s15*s23*s24*s34*s35-L2*s14*s23*s25*s34*s35+L2*s14*s23*s24*s35^2+L2*s13*s24^2*s35^2-L2*s15*s23*s25*s33*s44+L4*s14*s25*s33^2*s44-L4*s13*s25*s33*s34*s44+2*L2*s13*s23*s25*s35*s44-L4*s14*s23*s33*s35*s44-L4*s13*s24*s33*s35*s44+2*L4*s13*s23*s34*s35*s44+L2*s14*s23*s25*s33*s45+L2*s15*s23^2*s34*s45-L2*s13*s23*s25*s34*s45-L2*s14*s23^2*s35*s45-L2*s13*s23*s24*s35*s45+s15*s23*s33*s35*s44-2*s13*s23*s35^2*s44-s14*s25*s33^2*s45-s15*s23*s33*s34*s45+s13*s25*s33*s34*s45+s14*s23*s33*s35*s45+s13*s24*s33*s35*s45
_[28]=L1*s14^2*s25*s33*s34-L1*s13*s14*s25*s34^2-L1*s14^2*s24*s33*s35+L1*s13*s14*s24*s34*s35+L4*s14*s23*s24*s33*s44-L4*s13*s24^2*s33*s44-L1*s13*s14*s25*s33*s44-L4*s14*s23^2*s34*s44+L4*s13*s23*s24*s34*s44+L1*s13^2*s25*s34*s44+L1*s13*s14*s23*s35*s44-L1*s13^2*s24*s35*s44+L1*s13*s14*s24*s33*s45-L1*s13*s14*s23*s34*s45-s14*s24*s25*s33*s34+s14*s23*s25*s34^2+s14*s24^2*s33*s35-s14*s23*s24*s34*s35+s13*s24*s25*s33*s44-s13*s23*s25*s34*s44-s14*s23*s24*s33*s45+s14*s23^2*s34*s45


and it includes

Code:
_[2]=s15*s24*s33-s14*s25*s33-s15*s23*s34+s13*s25*s34+s14*s23*s35-s13*s24*s35


which is a completely irrelevant polynomial as it does not contain any L, hence I want to get rid of it.

Could it drop those s-polynomials during the calculation, or does it need to keep them to know which combinations of s-coefficients are zero to remove (s-polynomial)*(L polynomial) from the L-polynomials?

I thought eliminate would do the job, but it outputs nothing at all:
Code:
> eliminate(ideal(L1*L3*s13-L1*s14-L3*s23+s24, L1*L4*s14-L1*s15-L4*s24+s25, L2*L3*s23-L2*s24-L3*s33+s34, L2*L4*s24-L2*s25-L4*s34+s35, L3*L4*s34-L3*s35-L4*s44+s45), s12*s13*s14*s23*s24*s34);
_[1]=0



Also, I thought it might help to group the variables

Code:
> ring r = 0,(L1,L2,L3,L4,L5,L6,L7,L8,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),(dp(8), dp);
> groebner(ideal(L1*L3*s13-L1*s14-L3*s23+s24, L1*L4*s14-L1*s15-L4*s24+s25, L2*L3*s23-L2*s24-L3*s33+s34, L2*L4*s24-L2*s25-L4*s34+s35, L3*L4*s34-L3*s35-L4*s44+s45));
_[1]=s15*s24*s33-s14*s25*s33-s15*s23*s34+s13*s25*s34+s14*s23*s35-s13*s24*s35
_[2]=L3*s25*s33*s34-L3*s24*s33*s35-L4*s24*s33*s44+L4*s23*s34*s44-s25*s34^2+s24*s34*s35+s24*s33*s45-s23*s34*s45
_[3]=L3*s15*s33*s34-L3*s14*s33*s35-L4*s14*s33*s44+L4*s13*s34*s44-s15*s34^2+s14*s34*s35+s14*s33*s45-s13*s34*s45
_[4]=L3*s15*s23*s34-L3*s13*s25*s34-L3*s14*s23*s35+L3*s13*s24*s35-L4*s14*s23*s44+L4*s13*s24*s44-s15*s24*s34+s14*s25*s34+s14*s23*s45-s13*s24*s45
_[5]=L2*s24*s25*s34-L2*s24^2*s35-L2*s23*s25*s44+L2*s23*s24*s45+L4*s24*s33*s44-L4*s23*s34*s44+s23*s35*s44-s24*s33*s45
_[6]=L2*s14*s25^2*s33*s34+L2*s15*s23*s25*s34^2-L2*s13*s25^2*s34^2-L2*s15*s24^2*s33*s35-L2*s14*s23*s25*s34*s35+L2*s13*s24*s25*s34*s35-L2*s15*s23*s25*s33*s44+L2*s15*s23*s24*s33*s45+L4*s15*s24*s33^2*s44-L4*s15*s23*s33*s34*s44+s15*s23*s33*s35*s44-s15*s24*s33^2*s45
_[7]=L1*s14*s15*s34-L1*s14^2*s35-L1*s13*s15*s44+L1*s13*s14*s45+L4*s14*s23*s44-L4*s13*s24*s44-s14*s25*s34+s14*s24*s35+s13*s25*s44-s14*s23*s45
_[8]=L1*s14^2*s25*s33*s34+L1*s14*s15*s23*s34^2-L1*s13*s14*s25*s34^2-L1*s14^2*s24*s33*s35-L1*s14^2*s23*s34*s35+L1*s13*s14*s24*s34*s35-L1*s13*s15*s24*s33*s44+L1*s13*s14*s24*s33*s45+L4*s14*s23*s24*s33*s44-L4*s13*s24^2*s33*s44+s15*s24^2*s33*s34-2*s14*s24*s25*s33*s34-s15*s23*s24*s34^2+s13*s24*s25*s34^2+s14*s24^2*s33*s35+s14*s23*s24*s34*s35-s13*s24^2*s34*s35+s13*s24*s25*s33*s44-s14*s23*s24*s33*s45
_[9]=L4^2*s24*s33*s44-L4^2*s23*s34*s44+L4*s25*s34^2-L4*s24*s34*s35-L4*s25*s33*s44+L4*s23*s35*s44-L4*s24*s33*s45+L4*s23*s34*s45-s25*s34*s35+s24*s35^2+s25*s33*s45-s23*s35*s45
_[10]=L4^2*s14*s33*s44-L4^2*s13*s34*s44+L4*s15*s34^2-L4*s14*s34*s35-L4*s15*s33*s44+L4*s13*s35*s44-L4*s14*s33*s45+L4*s13*s34*s45-s15*s34*s35+s14*s35^2+s15*s33*s45-s13*s35*s45
_[11]=L4^2*s14*s23*s44-L4^2*s13*s24*s44+L4*s15*s24*s34-L4*s14*s25*s34-L4*s15*s23*s44+L4*s13*s25*s44-L4*s14*s23*s45+L4*s13*s24*s45-s15*s24*s35+s14*s25*s35+s15*s23*s45-s13*s25*s45
_[12]=L3*L4*s34-L3*s35-L4*s44+s45
_[13]=L3*L4*s24*s33-L3*L4*s23*s34-L3*s25*s33+L3*s23*s35+s25*s34-s24*s35
_[14]=L3*L4*s14*s33-L3*L4*s13*s34-L3*s15*s33+L3*s13*s35+s15*s34-s14*s35
_[15]=L3*L4*s14*s23-L3*L4*s13*s24-L3*s15*s23+L3*s13*s25+s15*s24-s14*s25
_[16]=L2*L4*s24-L2*s25-L4*s34+s35
_[17]=L2*L4*s23*s44-L3*L4*s33*s34-L2*s25*s34+L2*s24*s35-L2*s23*s45+L3*s33*s35
_[18]=L2*L4*s14*s25*s33+L2*L4*s15*s23*s34-L2*L4*s13*s25*s34-L2*L4*s14*s23*s35+L2*L4*s13*s24*s35-L2*s15*s25*s33-L4*s15*s33*s34+s15*s33*s35
_[19]=L1*L4*s14-L1*s15-L4*s24+s25
_[20]=L1*L4*s13*s44-L3*L4*s23*s34-L1*s15*s34+L1*s14*s35-L1*s13*s45+L3*s23*s35+s25*s34-s24*s35
_[21]=L2*L3*s23-L2*s24-L3*s33+s34
_[22]=L2*L3*s25*s34-L2*L3*s24*s35-L2*L4*s24*s44+L3*L4*s34^2+L2*s24*s45-L3*s34*s35
_[23]=L1*L3*s13-L1*s14-L3*s23+s24
_[24]=L1*L3*s15*s34-L1*L3*s14*s35-L1*L4*s14*s44+L3*L4*s24*s34+L1*s14*s45-L3*s25*s34
_[25]=L1*L2*s15*s24-L1*L2*s14*s25-L1*L4*s14*s34+L2*L4*s24^2+L1*s14*s35-L2*s24*s25
_[26]=L1*L2*s15*s23-L1*L2*s13*s25-L1*L4*s14*s33+L2*L4*s23*s24+L1*s13*s35-L2*s23*s25+L4*s24*s33-L4*s23*s34
_[27]=L1*L2*s14*s23-L1*L2*s13*s24-L1*L3*s13*s33+L2*L3*s23^2+L1*s13*s34-L2*s23*s24
_[28]=L1*L2*s14*s25*s34-L1*L2*s14*s24*s35-L1*L2*s13*s25*s44+L1*L2*s13*s24*s45+L1*L4*s14*s34^2-L1*L4*s13*s34*s44-L2*L4*s24^2*s34+L2*L4*s23*s24*s44-L1*s14*s34*s35+L1*s13*s35*s44+L2*s24^2*s35-L2*s23*s24*s45


but it does not. Not only is _[1]=s15*s24*s33-s14*s25*s33-s15*s23*s34+s13*s25*s34+s14*s23*s35-s13*s24*s35 still there (and moved to the beginning), the calculation has become 100 times slower. huh? ( for the input L1*L2*s12-L1*s13-L2*s22+s23, L1*L3*s13-L1*s14-L3*s23+s24, L1*L4*s14-L1*s15-L4*s24+s25, L2*L3*s23-L2*s24-L3*s33+s34, L2*L4*s24-L2*s25-L4*s34+s35, L3*L4*s34-L3*s35-L4*s44+s45 It takes all the time. The others are only like 50 % slower )


( to avoid an xy-problem: I do not actually need the entire Gröbner basis. I need to know if the equations contain a solution for each/some individual L, i.e. if the ideal contains a polynomial that depends only one a single L, like (s..) L1^2 + (s..) L1 + (s...) )


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hannes
 Post subject: Re: Gröbner basis of polynomials with unknown coefficients
PostPosted: Fri May 20, 2016 12:40 pm 

Joined: Wed May 25, 2005 4:16 pm
Posts: 275
Thre are two ways to compute Groebner bases with parameters:
- move the s.. to the coefficients (i.e. work with rational function in s...,
Code:
ring r=(0,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),(L1,L2,L3,L4,L5,L6,L7,L8),,dp;

- or (recommended), choose an ordering which separates the s.. from the L.. variables:
Code:
ring r = 0,(L1,L2,L3,L4,L5,L6,L7,L8,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),(dp(8),dp);


In this second, recommend case the Groebner basis contains
Code:
s15*s24*s33-s14*s25*s33-s15*s23*s34+s13*s25*s34+s14*s23*s35-s13*s24*s35=0

i.e. you have a condition on the parameters:: if this condition is not met the whole system cannot be fulfilled.

If you choose the first variant, the Groeber basis is simply
Code:
groebner(ideal(L1*L3*s13-L1*s14-L3*s23+s24, L1*L4*s14-L1*s15-L4*s24+s25, L2*L3*s23-L2*s24-L3*s33+s34, L2*L4*s24-L2*s25-L4*s34+s35, L3*L4*s34-L3*s35-L4*s44+s45));
_[1]=1

i.e. the system is (in general: with the exception of some condition on the parameters) not solvable.


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BeniBela
 Post subject: Re: Gröbner basis of polynomials with unknown coefficients
PostPosted: Fri May 20, 2016 1:13 pm 

Joined: Thu May 19, 2016 4:33 pm
Posts: 2
hannes wrote:
Thre are two ways to compute Groebner bases with parameters:
- move the s.. to the coefficients (i.e. work with rational function in s...,
Code:
ring r=(0,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),(L1,L2,L3,L4,L5,L6,L7,L8),,dp;




That looks interesting. One less comma, though

But I guess, it does not help if it turns into _[1]=1

hannes wrote:

- or (recommended), choose an ordering which separates the s.. from the L.. variables:
Code:
ring r = 0,(L1,L2,L3,L4,L5,L6,L7,L8,s11,s12,s13,s14,s15,s16,s17,s18,s21,s22,s23,s24,s25,s26,s27,s28,s31,s32,s33,s34,s35,s36,s37,s38,s41,s42,s43,s44,s45,s46,s47,s48,s51,s52,s53,s54,s55,s56,s57,s58,s61,s62,s63,s64,s65,s66,s67,s68,s71,s72,s73,s74,s75,s76,s77,s78,s81,s82,s83,s84,s85,s86,s87,s88),(dp(8),dp);



I tried that, but it becomes 100 times slower than the unseparated case.


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