Singular :: Post a reply

Forum FAQ

Forum Search

Back to Forum | View unanswered posts | View active topics

Gröbner basis of polynomials with unknown coefficients

Post a reply

Username:

Note:

If not registered, provide any username. For more comfort, register here.

Subject:

Message body:
Enter your message here, it may contain no more than 60000 characters.

Smilies

Font size:
	Font colour
[quote="BeniBela"]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: L1L3s13-L1s14-L3s23+s24 L1L4s14-L1s15-L4s24+s25 L2L3s23-L2s24-L3s33+s34 L2L4s24-L2s25-L4s34+s35 L3L4s34-L3s35-L4s44+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(L1L3s13-L1s14-L3s23+s24, L1L4s14-L1s15-L4s24+s25, L2L3s23-L2s24-L3s33+s34, L2L4s24-L2s25-L4s34+s35, L3L4s34-L3s35-L4s44+s45)); _[1]=L3L4s34-L3s35-L4s44+s45 _[2]=s15s24s33-s14s25s33-s15s23s34+s13s25s34+s14s23s35-s13s24s35 _[3]=L2L4s24-L2s25-L4s34+s35 _[4]=L2L3s23-L2s24-L3s33+s34 _[5]=L1L4s14-L1s15-L4s24+s25 _[6]=L1L3s13-L1s14-L3s23+s24 _[7]=L2L4s23s44-L2s25s34+L2s24s35-L4s33s44-L2s23s45+s33s45 _[8]=L1L4s13s44-L1s15s34+L1s14s35-L4s23s44-L1s13s45+s25s34-s24s35+s23s45 _[9]=L3s25s33s34-L3s24s33s35-L4s24s33s44+L4s23s34s44-s25s34^2+s24s34s35+s24s33s45-s23s34s45 _[10]=L3s15s33s34-L3s14s33s35-L4s14s33s44+L4s13s34s44-s15s34^2+s14s34s35+s14s33s45-s13s34s45 _[11]=L2s24s25s34-L2s24^2s35-L2s23s25s44+L4s24s33s44-L4s23s34s44+L2s23s24s45+s23s35s44-s24s33s45 _[12]=L2L3s25s34-L2L3s24s35-L2s25s44+L2s24s45+s35s44-s34s45 _[13]=L3s15s23s34-L3s13s25s34-L3s14s23s35+L3s13s24s35-L4s14s23s44+L4s13s24s44-s15s24s34+s14s25s34+s14s23s45-s13s24s45 _[14]=L1s14s15s34-L1s14^2s35-L1s13s15s44+L4s14s23s44-L4s13s24s44+L1s13s14s45-s14s25s34+s14s24s35+s13s25s44-s14s23s45 _[15]=L1L3s15s34-L1L3s14s35-L3s25s34+L3s24s35-L1s15s44+L1s14s45+s25s44-s24s45 _[16]=L3L4s24s33-L3s25s33-L4s23s44+s25s34-s24s35+s23s45 _[17]=L3L4s14s33-L3s15s33-L4s13s44+s15s34-s14s35+s13s45 _[18]=L1L2s15s24-L1L2s14s25-L1s15s34+L1s14s35+s25s34-s24s35 _[19]=L1L2s15s23-L1L2s13s25-L1s15s33+L1s13s35+s25s33-s23s35 _[20]=L3L4s14s23-L3L4s13s24-L3s15s23+L3s13s25+s15s24-s14s25 _[21]=L1L2s14s23-L1L2s13s24-L1s14s33+L1s13s34+s24s33-s23s34 _[22]=L4^2s24s33s44-L4^2s23s34s44+L4s25s34^2-L4s24s34s35-L4s25s33s44+L4s23s35s44-L4s24s33s45+L4s23s34s45-s25s34s35+s24s35^2+s25s33s45-s23s35s45 _[23]=L4^2s14s33s44-L4^2s13s34s44+L4s15s34^2-L4s14s34s35-L4s15s33s44+L4s13s35s44-L4s14s33s45+L4s13s34s45-s15s34s35+s14s35^2+s15s33s45-s13s35s45 _[24]=L4^2s14s23s44-L4^2s13s24s44+L4s15s24s34-L4s14s25s34-L4s15s23s44+L4s13s25s44-L4s14s23s45+L4s13s24s45-s15s24s35+s14s25s35+s15s23s45-s13s25s45 _[25]=L1L2s14s25s34-L1L2s14s24s35-L1L2s13s25s44+L1L2s13s24s45+L4s24s33s44-L4s23s34s44+L1s13s35s44-L1s13s34s45-s24s33s45+s23s34s45 _[26]=L2L4s14s25s33+L2L4s15s23s34-L2L4s13s25s34-L2L4s14s23s35-L2s15s25s33-L4s15s33s34+L2s13s25s35+L4s13s34s35+s15s33s35-s13s35^2 _[27]=L2s14s25^2s33s34+L2s15s23s25s34^2-L2s13s25^2s34^2-L2s14s24s25s33s35-L2s15s23s24s34s35-L2s14s23s25s34s35+L2s14s23s24s35^2+L2s13s24^2s35^2-L2s15s23s25s33s44+L4s14s25s33^2s44-L4s13s25s33s34s44+2L2s13s23s25s35s44-L4s14s23s33s35s44-L4s13s24s33s35s44+2L4s13s23s34s35s44+L2s14s23s25s33s45+L2s15s23^2s34s45-L2s13s23s25s34s45-L2s14s23^2s35s45-L2s13s23s24s35s45+s15s23s33s35s44-2s13s23s35^2s44-s14s25s33^2s45-s15s23s33s34s45+s13s25s33s34s45+s14s23s33s35s45+s13s24s33s35s45 _[28]=L1s14^2s25s33s34-L1s13s14s25s34^2-L1s14^2s24s33s35+L1s13s14s24s34s35+L4s14s23s24s33s44-L4s13s24^2s33s44-L1s13s14s25s33s44-L4s14s23^2s34s44+L4s13s23s24s34s44+L1s13^2s25s34s44+L1s13s14s23s35s44-L1s13^2s24s35s44+L1s13s14s24s33s45-L1s13s14s23s34s45-s14s24s25s33s34+s14s23s25s34^2+s14s24^2s33s35-s14s23s24s34s35+s13s24s25s33s44-s13s23s25s34s44-s14s23s24s33s45+s14s23^2s34s45 [/code] and it includes [code] _[2]=s15s24s33-s14s25s33-s15s23s34+s13s25s34+s14s23s35-s13s24s35 [/code] 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(L1L3s13-L1s14-L3s23+s24, L1L4s14-L1s15-L4s24+s25, L2L3s23-L2s24-L3s33+s34, L2L4s24-L2s25-L4s34+s35, L3L4s34-L3s35-L4s44+s45), s12s13s14s23s24s34); _[1]=0 [/code] 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(L1L3s13-L1s14-L3s23+s24, L1L4s14-L1s15-L4s24+s25, L2L3s23-L2s24-L3s33+s34, L2L4s24-L2s25-L4s34+s35, L3L4s34-L3s35-L4s44+s45)); _[1]=s15s24s33-s14s25s33-s15s23s34+s13s25s34+s14s23s35-s13s24s35 _[2]=L3s25s33s34-L3s24s33s35-L4s24s33s44+L4s23s34s44-s25s34^2+s24s34s35+s24s33s45-s23s34s45 _[3]=L3s15s33s34-L3s14s33s35-L4s14s33s44+L4s13s34s44-s15s34^2+s14s34s35+s14s33s45-s13s34s45 _[4]=L3s15s23s34-L3s13s25s34-L3s14s23s35+L3s13s24s35-L4s14s23s44+L4s13s24s44-s15s24s34+s14s25s34+s14s23s45-s13s24s45 _[5]=L2s24s25s34-L2s24^2s35-L2s23s25s44+L2s23s24s45+L4s24s33s44-L4s23s34s44+s23s35s44-s24s33s45 _[6]=L2s14s25^2s33s34+L2s15s23s25s34^2-L2s13s25^2s34^2-L2s15s24^2s33s35-L2s14s23s25s34s35+L2s13s24s25s34s35-L2s15s23s25s33s44+L2s15s23s24s33s45+L4s15s24s33^2s44-L4s15s23s33s34s44+s15s23s33s35s44-s15s24s33^2s45 _[7]=L1s14s15s34-L1s14^2s35-L1s13s15s44+L1s13s14s45+L4s14s23s44-L4s13s24s44-s14s25s34+s14s24s35+s13s25s44-s14s23s45 _[8]=L1s14^2s25s33s34+L1s14s15s23s34^2-L1s13s14s25s34^2-L1s14^2s24s33s35-L1s14^2s23s34s35+L1s13s14s24s34s35-L1s13s15s24s33s44+L1s13s14s24s33s45+L4s14s23s24s33s44-L4s13s24^2s33s44+s15s24^2s33s34-2s14s24s25s33s34-s15s23s24s34^2+s13s24s25s34^2+s14s24^2s33s35+s14s23s24s34s35-s13s24^2s34s35+s13s24s25s33s44-s14s23s24s33s45 _[9]=L4^2s24s33s44-L4^2s23s34s44+L4s25s34^2-L4s24s34s35-L4s25s33s44+L4s23s35s44-L4s24s33s45+L4s23s34s45-s25s34s35+s24s35^2+s25s33s45-s23s35s45 _[10]=L4^2s14s33s44-L4^2s13s34s44+L4s15s34^2-L4s14s34s35-L4s15s33s44+L4s13s35s44-L4s14s33s45+L4s13s34s45-s15s34s35+s14s35^2+s15s33s45-s13s35s45 _[11]=L4^2s14s23s44-L4^2s13s24s44+L4s15s24s34-L4s14s25s34-L4s15s23s44+L4s13s25s44-L4s14s23s45+L4s13s24s45-s15s24s35+s14s25s35+s15s23s45-s13s25s45 _[12]=L3L4s34-L3s35-L4s44+s45 _[13]=L3L4s24s33-L3L4s23s34-L3s25s33+L3s23s35+s25s34-s24s35 _[14]=L3L4s14s33-L3L4s13s34-L3s15s33+L3s13s35+s15s34-s14s35 _[15]=L3L4s14s23-L3L4s13s24-L3s15s23+L3s13s25+s15s24-s14s25 _[16]=L2L4s24-L2s25-L4s34+s35 _[17]=L2L4s23s44-L3L4s33s34-L2s25s34+L2s24s35-L2s23s45+L3s33s35 _[18]=L2L4s14s25s33+L2L4s15s23s34-L2L4s13s25s34-L2L4s14s23s35+L2L4s13s24s35-L2s15s25s33-L4s15s33s34+s15s33s35 _[19]=L1L4s14-L1s15-L4s24+s25 _[20]=L1L4s13s44-L3L4s23s34-L1s15s34+L1s14s35-L1s13s45+L3s23s35+s25s34-s24s35 _[21]=L2L3s23-L2s24-L3s33+s34 _[22]=L2L3s25s34-L2L3s24s35-L2L4s24s44+L3L4s34^2+L2s24s45-L3s34s35 _[23]=L1L3s13-L1s14-L3s23+s24 _[24]=L1L3s15s34-L1L3s14s35-L1L4s14s44+L3L4s24s34+L1s14s45-L3s25s34 _[25]=L1L2s15s24-L1L2s14s25-L1L4s14s34+L2L4s24^2+L1s14s35-L2s24s25 _[26]=L1L2s15s23-L1L2s13s25-L1L4s14s33+L2L4s23s24+L1s13s35-L2s23s25+L4s24s33-L4s23s34 _[27]=L1L2s14s23-L1L2s13s24-L1L3s13s33+L2L3s23^2+L1s13s34-L2s23s24 _[28]=L1L2s14s25s34-L1L2s14s24s35-L1L2s13s25s44+L1L2s13s24s45+L1L4s14s34^2-L1L4s13s34s44-L2L4s24^2s34+L2L4s23s24s44-L1s14s34s35+L1s13s35s44+L2s24^2s35-L2s23s24s45 [/code] but it does not. Not only is _[1]=s15s24s33-s14s25s33-s15s23s34+s13s25s34+s14s23s35-s13s24s35 still there (and moved to the beginning), the calculation has become 100 times slower. huh? ( for the input L1L2s12-L1s13-L2s22+s23, L1L3s13-L1s14-L3s23+s24, L1L4s14-L1s15-L4s24+s25, L2L3s23-L2s24-L3s33+s34, L2L4s24-L2s25-L4s34+s35, L3L4s34-L3s35-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...) )[/quote]

Options:

BBCode is ON

[img] is ON

[flash] is OFF

[url] is ON

Smilies are ON

	Disable BBCode
	Disable smilies
	Do not automatically parse URLs

Confirmation of post

To prevent automated posts the board requires you to enter a confirmation code. The code is displayed in the image you should see below. If you are visually impaired or cannot otherwise read this code please contact the %sBoard Administrator%s.

Confirmation code:
Enter the code exactly as it appears. All letters are case insensitive, there is no zero.

Topic review - Gröbner basis of polynomials with unknown coefficients

Author

Message

BeniBela

Post subject:

Re: Gröbner basis of polynomials with unknown coefficients

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.

Posted: Fri May 20, 2016 1:13 pm

hannes

Post subject:

Re: Gröbner basis of polynomials with unknown coefficients

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.

Posted: Fri May 20, 2016 12:40 pm

BeniBela

Post subject:

Gröbner basis of polynomials with unknown coefficients

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...) )

Posted: Thu May 19, 2016 6:25 pm

It is currently Fri May 13, 2022 10:56 am

Singular

Gröbner basis of polynomials with unknown coefficients

Main

Community

System

Misc