1

Given $n$ independent nonlinear equations in the $n+1$ variables $x_1,\dots,x_{n+1}$, is the solution always of dimension 1?

For example, if $n=1$, $x_1\sin(x_1+x_2)=0$ gives a curve in $\mathbb R^2$.

It is not clear to me whether this is something general are if it is only true under some conditions.

anderstood
  • 3,504

1 Answers1

2

Yes, this is something general; but no, this does not always happen. Counterexample: $x_1^2+x_2^2 = 0$, which is only true for $(x_1,x_2) = (0,0)$. This is because $(0,0)$ is a so-called critical point of the function $f(x,y) = x^2+y^2$. For more information, see Wikipedia: Level set. Having more than one (nonlinear) equation would then be equivalent to finding the intersection of different level sets.

Frits Veerman
  • 8,091
  • OK so if the the system of equations $f(x_1,\dots,x_n)$ has not critical point on an open set, then there is locally a variety of dimension 1? This seems to be the condition for the implicit function theorem to apply. – anderstood Oct 22 '15 at 16:38
  • That's exactly it. In other words, $f$ needs to be 'locally invertible'. – Frits Veerman Oct 22 '15 at 16:41