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I have recently watched this video by $3$Blue$1$Brown, regarding winding numbers and the fundamental theorem of algebra. I am trying to formalize the idea he shows in the video (starting at 21:26). Specifically, I am trying to connect his proof to a one I've seen in complex analysis classes:

Theorem. Fundamental theorem of algebra. Let $P(z)=z^d+a_1 z^{d-1}+a_2z^{d-2}+\ldots+a_{d-1}z+a_d\in \Bbb{C}[x]$ be a monic complex polynomial of degree $d$. So the sum of all orders of $P$'s zeroes is exactly $d$.

Proof. For a function $f$, denote the sum of orders of all its zeros as $N_f$. Let $Q(z)=z^d$. It is obvious that $N_Q=d$. Subtracting $Q$ from $P$ gives a polynomial of degree $d-1$, therefore $\frac{|P(z)-Q(z)|}{|Q(z)|}\to0$ as $z\to\infty$. Let $R_0$ be large enough so that for every $R>R_0$, every $|z|=R$ satisfies $|P(z)-Q(z)|\le |Q(z)|$. Using Rouché's theorem for the path $\gamma(t)=R e^{it}$ for $t\in[0,2\pi]$, we deduce that $N_P=N_Q=d$. $\tag*{$\blacksquare$}$

Understanding 3Blue1Brwon's Video

In his video, $3$B$1$B defines a winding number of a complex function on a path as the total number of times the image of the function on the path "goes through all hues" of the color map (with negative and positive directions), and explains why a nonzero winding number means that the function has a zero within the area of the path.

Qustion 1: What does he mean by this winding number? I.e., what is the formal definition of a "winding number" of a function? I am familiar with the definition $$\operatorname{Ind}_\gamma (a) := \frac{1}{2\pi i} \oint_\gamma \frac{\text{d}z}{z-a},$$ which defines the index of a (closed) path in respect to a point.

He then explains that the winding number of $Q(z)=z^d$ (in his video, $d=5$) around $0$ is $d$ (around some circular path). This I can intuitively understand - as $x$ goes along the unit circle, for example, the function $x^d$ goes around the circle $d$ times.

From 3B1B's video: the index of z^d

Qustion 2: How is it connected to $0$ being a zero of order $d$ of $Q(z)$?

Now, as $z\to\infty$, the leading term of $P(z)$ is the only significant one, so $3$B$1$B explains that in a large enough circle, the index of $P$ and $Q$ will be the same. This argument is quite similar to the one in the proof above -

Qustion 3: I assume that using Rouché's theorem "hides under the surface" some of the intuition in the proof. Is there a way of explicitly using this index equality of $P$ and $Q$ in a large enough circle formally in order to prove the fundamental theorem? Moreover, I know that paths that are homotopy equivalences, have the same index. Are $P$ and $Q$ in fact homotopy equivalences in the area outside the circle?

Summing up, I am looking for a more "mathematically complete" explanation of $3$B$1$B's argument, rigorous yet intuitive.

Edit:

Question 4: Rouché's theorem is proved using the so called argument principle, as Oliver Diaz's answer mentions. I assume that, intuitivly, we would have wanted to write $\int_\gamma \frac{f'(z)}{f(z)} \ \text{d}z=\ln(b)-\ln(a)$ (if $\gamma(0)=a$ and $\gamma(1)=b$), but that is of course nonsense, since the complex logarithm is not holomorphic in $[-\infty,0]$. That does remind me of the work done in a non-conservative field. Is this really the intuitive reason that the argument principle "works"?

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What Rouche's theorem says is that if two analytic functions $f,g$ are "relatively closed" (given by the condition that $|f-g|<f$ along some closed curve contained in the domain of analyticity) then both have the same nomeber of roots (according to multiplicity). So in the proof of the fundamental theorem of algebra, $Q(z)=z^d$ has $d$ roots ($0$ with multiplicity $d$) thus, by the fact that $P$ and $Q$ are relatively closed (along a large circular path for example), then $P$ has also $d$ roots inside the circular path. (some could have multiplicity larger than one, but when you add the multiplicities of each of the roots you get $d$)

The proof of Rouche's theorem is indeed based in the winding number of curves. In particular the following results


Lemma: Let $\gamma_0$ and $\gamma_1$ be closed paths in $\mathbb{C}$ parameterized by the interval $[a,b]$. If there is $\alpha\in\mathbb{C}$ such that $$ |\gamma_1(t)-\gamma_0(t)|<|\alpha-\gamma_0(t)|,\qquad a\leq t\leq b $$ then, $\operatorname{Ind}_{\gamma_0}(\alpha)=\operatorname{Ind}_{\gamma_1}(\alpha)$.

This result says roughly that if $\gamma_1$ and $\gamma_2$ are closed curves that are close relative to a point $\alpha$, then $\operatorname{Ind}_{\gamma_1}(\alpha)=\operatorname{Ind}_{\gamma_2}(\alpha)$.


The last ingredient is that the number of zeroes of the function can be directly connected to the winding (Index) number of a curve:

Theorem: Let $D\subset\mathbb{C}$ open and let $\gamma$ be a closed path (parameterized by the interval $[a,b]$) such that $\operatorname{Ind}_\gamma(z)=0$ for all $z\in D^c$. Suppose that $\operatorname{Ind}_\gamma(z)\in\{0,1\}$ for all $z\in D\setminus\gamma^*$. If $f$ is an analytic function on $D$ and If $f(\gamma(t))\neq0$ for all $a\leq t\leq b$, then the number of zeroes $N_f$ of $f$ in $D_1=\{z\in D: \operatorname{Ind}_\gamma(z)=1\}$, counted according to their multiplicity, is finite and \begin{aligned} N_f=\frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,dz=\operatorname{Ind}_{\gamma_f}(0) \end{aligned} where $\gamma_f:=f\circ\gamma$.


I hope this give you more context to the intuition you got from the video you mentioned.

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  • $\begingroup$ This does not quite answer my question. I have seen this proof of Rouche's theorem (and of the so-called argument principle). This still remains, in my opinion, in the "background" of the proof, and I do not see how the index is explicitly used in 3B1B's "proof". I understand the formal mathematics, but I lack the intuition (and connection to the video). $\endgroup$ – Roy Sht Jun 18 at 0:48

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