Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 3 / 14H (Complex Analysis)

Let $p$ first be an arbitrary degree-$n$ polynomial with leading coefficient $a_n\ne 0$. Choose $R$ so large that on $|z|=R$,The homotopy $a_n{z}^n+s[p(z)-a_n{z}^n]$, $0\le s\le 1$, never meets zero on that circle. Hence the image curve $p(R{e}^{i\theta })$ has winding number $n$, the same as $a_n{R}^n{e}^{in\theta }$. The argument principle counts precisely $n$ zeros inside, with multiplicity, since there are no poles. Choose $R$ large enough that the same leading-term domination holds for every $|z|\ge R$; the lower-term ratios decrease with radius, so this also excludes all exterior zeros. This proves the fundamental theorem of algebra with its multiplicity count. A degree-zero nonzero polynomial has no zeros.

Now use the specified bound for the monic quartic. Compare $\Gamma$ with the closed curveIts first piece travels once counterclockwise around a circle, and its second stays at the endpoint, so its winding number is one. On the first piece, $|\Gamma -{\gamma }_0|<R^4$. On the second, both $p(R)$ and $p(iR)$ differ from $R^4$ by less than $R^4$; every convex combination of those errors also has modulus less than $R^4$. Thus the quoted winding number stability result gives

For real coefficients define $q(y)=y^4-b{y}^2+d$. It has no real zeros and positive leading coefficient, so $q(y)>0$ for every real $y$, including $d=q(0)>0$. Thereforelies strictly in the right half-plane for every real $y$. Also $p(x)>0$ for $x\ge 0$: there are no zeros there, and the polynomial is positive for sufficiently large positive $x$. These facts exclude roots on the positive real and imaginary axes.

Consider the positively oriented quarter-disk boundary: first the arc from $R$ to $iR$, then the imaginary axis to zero, then the real axis back to $R$. Under $p$, its last two pieces form a path from $p(iR)$ to $p(R)$ entirely within the right half-plane. The chord closing $\Gamma$ also lies there. Since this half-plane is convex and excludes zero, deforming one closing path into the other changes no winding number. Thus the full image of the quarter-disk boundary has winding number one. By the argument principle there is exactly one root inside that quarter-disk, counted with multiplicity.

Finally, all polynomial roots lie in $|z|<R$ under the given strict coefficient bound: for $|z|\ge R$, the sum of lower terms divided by $|z{|}^4$ is smaller than one. HenceIts multiplicity is one. This is the quarter-sector winding test for a real quartic; it accounts for the axis exclusions and does not replace the sector boundary by a chord without justifying the deformation.