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check solutions in systems of polynomial equations
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1757		 Page 1 of 1
Author:  madalina [ Mon Jul 20, 2009 3:20 pm ]
Post subject:  check solutions in systems of polynomial equations
I have the following problem that I want to solve with Singular.
I have an original system with 2 variables and 3 unknowns:
f0=f(x,y)=0;
f1=dx(f)=0;
f2=dy(y)=0;
where dx, dy represent the partial derivatives w.r.t x and y.

I am constructing an equivalent system:
g1=a*f(x,y)+b*dx(f)+c*dy(f)=0
g2=d*f(x,y)+e*dx(f)+g*dy(f)=0,
where a,b,..g are integers.
I am computing the solutions of the equivalent system using "solve" command from Singular.
Then I want to take this solutions, round them and check which of the rounding solutions verify my original system.
Everything is okay in the source code until I want to substitute the rounding solutions into the original system.
Since the lists of solutions SOL[i][1] is defined in a ring, and since subst is defined in another ring, I cannot seem to find a way to substitute the solutions into the original system.

I used the following commands for solving the equivalent system:
Code:
ideal m=g1,g2;
ideal n = groebner(m);
setring r;
def T=solve(n,30,1,"nodisplay");


and then the following commands for substituting the solutions in the original system:
Code:
int k;
for (k=1; k<=size(SOL); k++){
      l[k];
      number solAx = l[k][1];
      number solAy = l[k][2];
           list lsols= list(round(solAx),round(solAy));
      lsols[1];lsols[2];
      //setring r;
      poly f0=-1 + 12*x + 6*x2 + x3 + 6*y - y2;
      poly f1=diff(f0,x);
      poly f2=diff(f0,y);
                setring r;
      subst(f0,lsols[1],y,lsols[2]);
      subst(f1,x,lsols[1],y,lsols[2]);
      subst(f2,x,lsols[1],y,lsols[2]);
                setring T;
   }

Still I get this error message:
`lsols` is not defined
when compiling since lsols is not recognized in the ring r.
I understand perfectly the error, still since I am a beginner in Singular, I don't seem to understand how should I solve this problem or want I should use from Singular to solve it.

Thank you in advance for any suggestions,
madalina
Author:  Guest [ Tue Jul 21, 2009 1:18 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
I've solve the problem.
I used "imap" function to map the approximate solutions from the ring "T" to the ring "r", and then I substitute the map solutions into the original system.
I get exactly what I need.

[code]
list newl=imap(T,lsols);
subst(f0,x,newl[1],y,newl[2]);
[code]

madalina.
Author:  Guest [ Tue Jul 21, 2009 1:19 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
[quote="Guest"]I've solve the problem.
I used "imap" function to map the approximate solutions from the ring "T" to the ring "r", and then I substitute the map solutions into the original system.
I get exactly what I need.

Code:
list newl=imap(T,lsols);
subst(f0,x,newl[1],y,newl[2]);


madalina.
Author:  Guest [ Tue Jul 21, 2009 1:19 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
I've solve the problem.
I used "imap" function to map the approximate solutions from the ring "T" to the ring "r", and then I substitute the map solutions into the original system.
I get exactly what I need.

Code:
list newl=imap(T,lsols);
subst(f0,x,newl[1],y,newl[2]);


madalina.[/quote]
Author:  Guest [ Tue Jul 21, 2009 1:20 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
I've solve the problem.
I used "imap" function to map the approximate solutions from the ring "T" to the ring "r", and then I substitute the map solutions into the original system.
I get exactly what I need.

Code:
list newl=imap(T,lsols);
subst(f0,x,newl[1],y,newl[2]);


madalina.
Author:  Guest [ Tue Jul 21, 2009 1:20 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
I've solved the problem.
I used "imap" function to map the approximate solutions from the ring "T" to the ring "r", and then I substitute the map solutions into the original system.
I get exactly what I need.

Code:
list newl=imap(T,lsols);
subst(f0,x,newl[1],y,newl[2]);


madalina.
Author:  seelisch [ Tue Jul 21, 2009 6:48 pm ]
Post subject:  Re: check solutions in systems of polynomial equations
Madalina,

you may just as well map your polynomials f0, f1, f2 (which live in your initial ring r, say) into the ring T (where SOL also lives). This may in general be better for checking the solutions, as T has a much higher precision, at least in the setup you posted.
By the way, for mapping polynomials, you may also use the more handy command 'fetch'.

Regards,
Frank
Author:  seelisch [ Fri Jul 24, 2009 10:30 am ]
Post subject:  Re: check solutions in systems of polynomial equations
Madalina,

let me correct my last comment. I was mixing things up a bit. Of course, you should be computing over the rationals in your ring r, i.e. exact. Thus mapping back your 30-digit-precision complex solutions from T to r will result in rationals which exactly preserve this precision. Therefore, it will NOT be more exact to check the solutions in T, contrary to my initial comment.

As to 'fetch', there's actually not much of a difference. In general, 'fetch' maps the i-th ring variable of the first ring to the i-th of the second, whereas 'imap' maps between variables with the same name. Depending on the context, the one or the other may be more useful.

Regards,
Frank
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