Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$
The point $x^* = (0,0,0)^T$ is obviously a solution to this system. Compute the number of solution branches through this point and the tangent directions along all solution branches.
I don't really have an idea of how to solve this question. I tried plotting this, but I don't seem to have any branch points?
On the other hand I could calculate the Jacobian: $$DF(x) = \begin{pmatrix}1+7x^6 & 1+7y^6 & 1+9z^8\\1 & -1 & 1+\sin z \end{pmatrix}$$ Which evaluates to $$DF(x^*) = \begin{pmatrix}1 & 1 & 1\\1 & -1 & 1 \end{pmatrix}$$ Should I look for a right zero vector? Somehting like $(1,0,-1)^T$