How to prove there exists a solution? Guillemin Pollack Prove there exists a complex number $z$ such that 
$$
z^7+\cos(|z^2|)(1+93z^4)=0.
$$
(For heaven's sake don't try to compute it!)

As pointed out in the comments, not only is there a complex number satisfying the equation; there is at least one real number such that
$$
f(z)=z^7 + \cos(z^2)\left(1+93z^4\right)=0.
$$
To see this, just note that $f(z)$ is continuous, and negative for small enough real $z$, and positive for large enough real $z$; then apply the intermediate value theorem.
