How eliminate variables using gb?

[Eduardo Mendes]

Hello

I am a newbie and wonder how Macaulay2 can be used to eliminate variables in a polynomial ideal. 

Here it is an example of what I am trying to accomplish.

R=QQ[x,y,z,x0,x1,x2,x3,z0,z1,z2,z3]

S=QQ[a,b,c,d,e,f,g,h,i,j,k,l,m]

T=R**S

I=ideal(-x + x0,x1 - a*y,x2 - a*(b*x + (c + d*y)*z), x3 - a*(a*b*y + d*z*(b*x + (c + d*y)*z) +

            c*(m + j*x**2 + f*y + k*y*y + x*(e + h*y) + (g +i*x)*z + l*z*z) + d*y*(m + j*x*x + f*y + k*y*y +

      x*(e + h*y) + (g + i*x)*z + l*z*z)))

G = gb(I);

and

gens G

prints an output with several components. mingens returns less components and one of them seems to be function of only x0, x1, x2 and x3

 (exactly what I need).  

Is there an easy way for Macaulay2 to return only the polynomial in x0,x1,x2 and x3?  

How to check the length of G?

How to access each of the components of G separately?  

Many thanks

Ed

[Thomas Kahle]

Hi,

Macaulay2 has built in functionality for elimination. You can start
reading here:

http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9/share/doc/Macaulay2/Macaulay2Doc/html/_elimination_spof_spvariables.html

and this is the eliminate command:

http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9/share/doc/Macaulay2/Elimination/html/_eliminate.html

Cheers,
Thomas

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[Eduardo Mendes]

Hi Thomas

Thank you ever so much.

I have managed to change the example in the second link to my needs.  It worked for one of the problems I am dealing with but not for the other.   Increase MAXHINCR or MAX_HEAP_SECTS.

I have checked the forum for a solution and it seems that I have to compile macaulay2 from scratch.  

Cheers

Ed

[Michael Stillman]

No, check a bit more.  You can also set the initial heap size as an environment variable.  I posted such a solution fairly recently (I can search and find it if needed).

Also, there are a number of things that can be done to improve the space/time of this operation, at least in certain cases.  If you post the example, or email it to me, I can take a look too.  

Best,

Mike

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[Eduardo Mendes]

Hi Mike

Many thanks.

If it is  not too much trouble, could you send me the link to the solution or the solution itself, please?  

I have just sent you an e-mail with the ideal (a bit too long for posting it here).

Cheers

Ed