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Solving a polynomial system with parameters
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1582		 Page 1 of 1
Author:  ssegvic [ Wed Dec 06, 2006 11:41 am ]
Post subject:  Solving a polynomial system with parameters
Hello,

I am trying t use Singular 3.0.2 to symbolically solve
a polynomial system of equations with parameters.
As a guide, I used the following example:
doc/examples/Solving_systems_of_polynomial_equations.sing

Unfortunately, the last call (triangMH) of the program given below
never ends.

Any help or suggestion would be greatly appreciated :-)

Sinisa

//I want to solve for l1,l2,l3,l4 in terms of a,b,c,d,d,e,f
LIB "solve.lib";
ring r=(0,a,b,c,d,e,f),(l1,l2,l3,l4),dp;
poly f1=2*l1*b+2*l2*e+2*a*l3+2*l4*l1;
poly f2=2*l1*e+2*l2*d+2*c*l3+2*l4*l2;
poly f3=2*l1*a+2*l2*c+2*f*l3;
poly f4=l1^2+l2^2-1;

ideal i=f1,f2,f3,f4;
ideal si=std(i);

option(redSB);
ideal maxI=maxideal(1);
ideal j=sat(si,maxI)[1];
vdim(j);

ring R=(0,a,b,c,d,e,f),(l1,l2,l3,l4),dp;
ideal j=fglm(r,j);
list L=triangMH(j);
Author:  greuel [ Sat Dec 09, 2006 1:37 am ]
Post subject: 
Hello,
if you change in the definition of R the ordering to lp (lex), then
ideal j=fglm(r,j); solves the system already. You can apply
L=triangMH(j); but this does not change anything.
Thus, the following works:

ring R=(0,a,b,c,d,e,f),(l1,l2,l3,l4),lp;
ideal j=fglm(r,j);
j;

/*You get a triangular system in which l4 satisfies a quadratic equation
j[1]=l4^2+(-a^2+b*f-c^2+d*f)/(f)*l4+(-a^2*d+2*a*c*e-b*c^2+b*d*f-e^2*f)/(f)
j[2]=l3^2+... (something involving l4)
etc ...
Symbolically, this is the best you can hope for.
*/
Author:  ssegvic [ Mon Dec 11, 2006 2:47 pm ]
Post subject:  A real problem
greuel wrote:
Hello,
if you change in the definition of R the ordering to lp (lex), then
ideal j=fglm(r,j); solves the system already.


Thanks!

However, that was a toy example, just for warming up :-)
One of our real problems is:

LIB "solve.lib";
ring r=(0,f1,f2,f3,f4,f5,f6,f7,f8,f9),(x1,x2,y1,y2,la,e),dp;

poly fa=2*x1+la*(f1*x2+f2*y2+f3);
poly fb=2*x2+la*(x1*f1+y1*f4+f7);
poly fc=2*y1+la*(f4*x2+y2*f5+f6);
poly fd=2*y2+la*(x1*f2+y1*f5+f8);
poly fe=(x1*f1+y1*f4+f7)*x2+(x1*f2+y1*f5+f8)*y2+x1*f3+y1*f6+f9;
poly ff=x1^2+x2^2+y1^2+y2^2-e;

ideal i=fa,fb,fc,fd,fe,ff;
ideal si=std(i);

The last call was entered two hours ago,
and there are no results yet.

Is there a better way to formulate the above?

Sinisa
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