21 January 2009

The fundamental theorem of algebra

The simplest proof of the fundamental theorem of algebra (every polynomial has a complex root), to the best of my knowledge, is the one by Anton R. Schep, just published in the American Mathematical Monthly, Volume 116, Number 1, January 2009 , pp. 67-68(2). It's worth knowing and so here it goes:

Suppose that p(z) is a polynomial in the complex variable z and that p(z) is never zero. Then 1/p(z) is an entire (analytic everywhere) function. By Cauchy's theorem,
I(r) := ∫ (1/zp(z)) dz = 2π i/p(0),
where the integral I(r) is taken over a circle of radius r and centered at the origin. But
|I(r)| ≤ ∫ (1/|z| |p(z)| ) |dz| ≤ (1/r m(r)) ∫ |dz|,
where m(r) is the smallest value of |p(z)| when z ranges on the circle, while the latter integral in the display above is, clearly, the circumference of the circle, i.e. 2 π r. Combining the above we see that
m(r) ≤ p(0).
But this can't be true for all r because, as r → ∞, we have m(r) → ∞. QED

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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.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant