1
$\begingroup$

I can not find anywhere whether Mathematica finds ALL the complex solutions to complex polynomial systems or just some of them.

For example, I typed

Solve[{x y^2 - y z^2 + x^5 + a x^2 z^2 == 0, D[x y^2 - y z^2 + x^5 + a x^2 z^2, x] == 0, D[x y^2 - y z^2 + x^5 + a x^2 z^2, y] == 0, D[x y^2 - y z^2 + x^5 + a x^2 z^2, z] == 0}, {x, y, z, a}]

and I got (0,0,0) and probably two singular curves, but there's a notification saying "Solve: Equations may not give solutions for all "solve" variables." What does that mean?

Thank you.

$\endgroup$
  • 2
    $\begingroup$ Your output will show solutions where (for example) y and z are functions of x which means there are an infinite number of solutions. So the 4 equations you list don't give you finite set of solutions. The (0,0,0) solution allows a to take on any value. Using Reduce in place of Solve might help. $\endgroup$ – JimB Apr 3 '17 at 18:05
  • $\begingroup$ I see, thank you @Jim, very useful comments. However, I also meant more in general: can Mathematica find literally all the solutions of a polynomial system over complex numbers? I mean can I trust that there's no other solution that Mathematica might miss? $\endgroup$ – Kristina Apr 4 '17 at 15:30
1
$\begingroup$
eqns = {
   x y^2 - y z^2 + x^5 + a x^2 z^2 == 0,
   D[x y^2 - y z^2 + x^5 + a x^2 z^2, x] == 0,
   D[x y^2 - y z^2 + x^5 + a x^2 z^2, y] == 0,
   D[x y^2 - y z^2 + x^5 + a x^2 z^2, z] == 0};

vars = {x, y, z, a};

soln = Solve[eqns, vars]

enter image description here

Verifying the solutions

And @@ (And @@ eqns /. soln)

(*  True  *)

Each solution leaves a variable unspecified, i.e., that variable can have any value (real or complex) and still satisfy the equations.

Map[First, soln, {2}]

(*  {{x, y, z}, {y, z, a}, {y, z, a}, {y, z, a}, {y, z, a}}  *)

Since a "solve" variable is not specified, the warning is issued.

$\endgroup$
1
$\begingroup$

Let us consider

Reduce[{x y^2 - y z^2 + x^5 + a x^2 z^2 == 0,D[x y^2 - y z^2 + x^5 + a x^2 z^2, x] == 0,
D[x y^2 - y z^2 + x^5 + a x^2 z^2, y] == 0,D[x y^2 - y z^2 + x^5 + a x^2 z^2, z] == 0},
{x, y, z, a}]

(x==0&&y==0&&z==0)||((y==-I x^2||y==I x^2)&&(z==-Sqrt[2] Sqrt[x] Sqrt[y]||z==Sqrt[2] Sqrt[x] Sqrt[y])&&x!=0&&a==y/x^2)

i.e. the system under consideration has an infinite set of its solutions. Hope this sheds light on the warning.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.