The fundamental theorem of algebra states that:
Every non-zero, single-variable, degree $n\,$ polynomial with complex coefficients has, counted with multiplicity, exactly $n\,$ complex roots.
The term I want to understand is multiplicity. Now, I already know the following:
The multiplicity of a root $x_0$ of a polynomial equation $p(x) = 0\,$ tells us how many times the factor $(x – x_0)\,$ divides the polynomial $p(x)$. That's fine.
But what I want to have, is an intuition. The definition of a root is: $x_0\,$ is a root of $p(x) = 0\,$ if and only if $p(x_0) = 0$. So, I don't understand how a root could have a property called multiplicity. For example, the multiplicity of a root $x_0$ cannot be the number of times $x_0\,$ solves the equation $p(x) = 0\,$ because a root cannot really solve an equation more than once.
Nor does it make sense as the number of times the graph of the function $y = p(x)\,$ meets the $x$-axis at the $x$-coordinate that is $x_0$, because they meet at a given point only once [because of the definition of a function]. But I do know of the geometric intuition: If multiplicity is odd, then the axis is crossed, otherwise it touches and comes back, and the higher the multiplicity, the closer $p(x)\,$ stays near the $x$-axis in the neighborhood of $x = x _0$, and so on.
What I want to have is an intuition, an answer to the question:
What is the multiplicity of a root of a polynomial equation the multiplicity of?
Can somebody plz help me on this one?
NOTE: Here, I am using multiplicity the same way I would use the term frequency. Plz correct me if this usage is wrong.
Best Answer
The fundamental theorem of algebra is stated in the complex domain ($\mathbb C$). This reminded us that maybe we could have an answer with complex ($\mathbb C$) viewpoint.
Let $f(z)$ be a polynomial over $\mathbb C$, and $u\in\mathbb C$ be a root of $f(z)$ of multiplicity $m$. This is expressed by an equation: $$f(z)=(z-u)^m g(z)$$ where $g(z)$ is a polynomial that does not have $u$ as its root.
Clearly $f$ brings $u$ to $0$, from one complex plane to another. This characterizes the notion of root geometrically:
Since $f$ is continuous, points near $u$ are mapped to some points near $0$. By complex analysis, a small loop around $u$ should be mapped to another loop winding $m$ times around the origin. The following picture shows the case when $m=2$:
Example:
For every complex number $a$ we can assign a color for it according to its angle $\operatorname{arg} a$. The brighter color means that the angle is closer to $0^\circ$:
Let $f(z)=(z+1)^3(z-2)$. For every point $z$ on the complex plane we paint the corresponding color for $\operatorname{arg}f(z)$:
The horizontal axis is real and the vertical one is imaginary. The small green squares indicate the unit length.
When we travel along a path $\gamma$ around $z=-1$, which is the root of multiplicity $3$, the color changes (white to black) three times. This shows that the corresponding path $f(\gamma)$ loops three times around the origin of the codomain. The discussion of the other root $z=2$ is similar.
This characterizes the notion of multiplicity geometrically, so I will try to prove this property (or, at least, give an idea of how it happens, because I'm not really familiar with complex analysis). But so far you can just think that the multiplicity is (or coincides with) how many times the color changes. That's the viewpoint that I wanted to provide.
(In fact, you can count the change of the color around a circle that contains all roots. The result coincides with the degree of $f$. This is related to a proof of the fundamental theorem of algebra.)
Proof (or explanation):
I will work on this statement:
Since $\gamma$ does not contain other root of $f$, $\gamma$ contains no root of $g$. This means if we let the radius $r$ tend to zero, then $\gamma$ shrinks to $u$ gradually without passing through any root of $g$. Thus the corresponding curve $g\circ \gamma$ can shrink to a point without passing through zero.
(I implicitly used the continuity of $g$.)
Now, let's simplify $\gamma_2$: $$\begin{aligned} \gamma_2(\theta)=f(u+re^{i\theta}) &= ((u+re^{i\theta})-u)^mg(u+re^{i\theta})\\[0.7em] &= re^{im\theta}g(u+re^{i\theta}). \end{aligned}$$
Since $g(u+re^{i\theta})$ can shrink continuously to a point without passing through the origin, the net change of angle of the loop $g(u+re^{i\theta})$ w.r.t the origin is zero.
(I implicitly use some property of the homotopy)
So the net change of angle of $\gamma_2$ is only caused by $re^{im\theta}$, which winds $m$ times around the origin.