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.
y
andz
are functions ofx
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 allowsa
to take on any value. UsingReduce
in place ofSolve
might help. $\endgroup$