How to prove there exists a solution? Guillemin Pollack

Question

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!)

Answer 1

If you look at a circle of radius $2$, then the argument of $f(z) = z^7 + \cos(|z|^2)(1+93z^4)$, which happens to be $z^7 + \cos {49}(1+93z^4)$, makes $7$ loops around the circle (using Rouché's theorem, for example).

Meanwhile, if you look at a small circle around $0$, the argument of $f(z)$ won't make any loop (because $f(0) = 1$ and $f$ is continuous)

So while the radius goes from $2$ to $0$, the number of loops has to jump despite $f$ being continuous, and this can only happen when $f$ has a zero.

Answer 2

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.