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Groebner bases with vs without parameters
http://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2670
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Author:  esche [ Tue Sep 19, 2017 2:44 pm ]
Post subject:  Groebner bases with vs without parameters

Dear Singular Forum,

I would like to perform polynomial division in a polynomial ring with parameters. Here is an example that I have in mind:
Code:
ring r=(0, y(1)),(z(1), z(2), z(3), z(4), z(5)),dp;
ideal f = z(1)*z(2)*z(3)*z(4)*z(5)-y(1)*z(2)^5+z(3)^5+z(4)^5+z(5)^5+z(1)^5;
ideal j = jacob(f);
ideal gb = groebner(j,"par2var");
poly p = 24*y(1)^4*z(2)^20;
list q = division(p,j);

This yields (note result of the division is zero and there are rational expressions in the parameter)
Code:
q;
[1]:
   _[1,1]=(24*y(1))/(78125*y(1)-25)*z(1)^2*z(2)^6*z(3)*z(4)*z(5)^6
   _[2,1]=(-24*y(1)^3)/5*z(2)^16+(24*y(1)^2)/5*z(2)^11*z(5)^5+24/(390625*y(1)-125)*z(1)^2*z(2)^3*z(3)^2*z(4)^2*z(5)^7+(-24*y(1))/5*z(2)^6*z(5)^10+24/5*z(2)*z(5)^15
   _[3,1]=(-24*y(1))/(15625*y(1)-5)*z(1)^5*z(2)^5*z(3)*z(5)^5
   _[4,1]=(24*y(1))/(3125*y(1)-1)*z(1)^4*z(2)^4*z(3)^4*z(5)^4
   _[5,1]=(24*y(1)^3)/5*z(2)^15*z(5)+(-120*y(1))/(3125*y(1)-1)*z(1)^3*z(2)^3*z(3)^3*z(4)^3*z(5)^4+(-24*y(1)^2)/5*z(2)^10*z(5)^6+24/125*z(1)^2*z(2)^2*z(3)^2*z(4)^2*z(5)^8+(24*y(1))/5*z(2)^5*z(5)^11-24/25*z(1)*z(2)*z(3)*z(4)*z(5)^12
[2]:
   _[1]=0
[3]:
   _[1,1]=1

For efficiency reasons I would like to perform the same computation treating y(1) as a ring variable. I observed that with several parameters treated as ring variables the Groebner basis computation is much faster than if they are treated as parameters. So here is the same example as above
Code:
ring r = 0, (y(1), z(1), z(2), z(3), z(4), z(5)), dp;
ideal f = z(1)*z(2)*z(3)*z(4)*z(5)-y(1)*z(2)^5+z(3)^5+z(4)^5+z(5)^5+z(1)^5;
int i;
ideal j=diff(f,z(1));
for(i=2;i<=5;i=i+1){j=j,diff(f,z(i));};
ideal gb = groebner(j);
poly p = 24*y(1)^4*z(2)^20;
list q = division(p,j);

The result, however, is very different. In particular the result of the division is not zero anymore.
Code:
q;
[1]:
   _[1,1]=0
   _[2,1]=-24/5*y(1)^3*z(2)^16-24/25*y(1)^2*z(1)*z(2)^12*z(3)*z(4)*z(5)-24/5*y(1)*z(2)^6*z(5)^10+24/5*z(2)*z(5)^15
   _[3,1]=0
   _[4,1]=0
   _[5,1]=24/25*y(1)^2*z(1)*z(2)^11*z(3)*z(4)*z(5)^2-24/5*y(1)^2*z(2)^10*z(5)^6+24/5*y(1)*z(2)^5*z(5)^11-24/5*z(5)^16
[2]:
   _[1]=24*z(5)^20
[3]:
   _[1,1]=1

How do I have to modify the code in order to reproduce the correct result of the division in the ring with parameter ?

Thanks for the help in advance.

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