Number of roots in the first quadrant I want to find how many roots of the equation $z^4+z^3+1=0$ lies in the first quadrant.
Using Rouche's Theorem how to find ?

Look at the family of polynomials $z^4+tz^3+1$. For $t=0$ we know the solutions $z=\sqrt{\frac12}(\pm1\pm i)$ which has one root per quadrant. We additionally know that the set of roots is continuous in the coefficients of the polynomial.
Now if changing $t$ from $0$ to $1$ were to change the number of roots in the first quadrant, one of the other roots would have to pass the positive $x$ or $y$ axis. However, on the positive $x$ axis the real part $1+x^3+x^4$ and on the $y$ axis the real part $1+y^4$ are never zero. Thus $$|z^4+tz^3+1|\ge |z^4+1|-t|z|^3>0$$ on the boundary of the first quadrant for $t\in [0,1]$. There is no change in the number of roots over this homotopy.

roots of $z^4+tz^3+1$ for red: $t=0$ over blue to green: $t=1$
