3 April 2011

A new very short proof of the fundamental theorem of algebra

I've always been intrigued by the fundamental theorem of algebra (every nonconstant polynomial with complex coefficients has a root), not least because I don't know any proof which uses algebra only. Earlier, I posted an easy proof in this blog, one that uses Cauchy's theorem.There is a recent proof (Oswaldo Rio Branco de Oliveira, Mathem. Intellig., March 2011), which is almost trivial. It goes as follows (and this is a chance for me to see if the embedded LaTeX script works...):

Let $P(z)$ be a polynomial of degree $n$. Since $|P(z)|$ is a nonnegative continuous function, tending to $\infty$ as $|z|$ tends to $\infty$, it has a minimum at some point $z_0$:
$|P(z)| \ge |P(z_0)|$, for all $z$.
By division of $P(z)-P(z_0)$ by $z-z_0$, write
$P(z) = P(z_0) + (z-z_0)^k Q(z-z_0),$
where $Q(0) \not = 0$. Since $P(z)$ is nonconstant, the integer $k$ is $\ge 1$.
$|P(z_0) + (z-z_0)^k Q(z-z_0)|^2 \ge |P(z_0)|^2$, for all $z$,
and, expanding the square,
$|z-z_0|^{2k} |Q(z-z_0)|^2 + 2 \Re \{ (z-z_0)^k Q(z-z_0) \overline{P(z_0)}\} \ge 0$, for all $z$.
Let $z=z_0 + r e^{i \theta}$, divide by $r^k$, and let $r$ tend to $0$. We obtain
$\Re \{ e^{i k \theta}  Q(0) \overline{P(z_0)}\} \ge 0$,  for all real $\theta$.
It is an easy exercise in algebra that, if $\alpha$ is a complex number such that $\Re \{ e^{i k \theta} \alpha\} \ge 0$ for all $\theta$, then $\alpha=0$. Hence $Q(0) \overline{P(z_0)}=0$. Since $Q(0) \neq 0$, we obtain $P(z_0)=0$.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
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