= Brownian proof of the fundamental theorem of algebra
{c}
If a nonconstant complex polynomial $P$ had no zeros, its reciprocal $g=1/P$ would be an entire <holomorphic function>, bounded and tending to zero at infinity. Its real and imaginary parts are <harmonic functions>. Thus $g(z+B_t)$, for <planar Brownian motion> started at zero, is a bounded complex <martingale> with expectation $g(z)$. For fixed $R$, the Gaussian density bounds $\mathbb P(|z+B_t|\leq R)$ by $R^2/(2t)$. Outside a sufficiently large disk, $|g|\leq\varepsilon$. Hence $|g(z)|\leq\varepsilon+\|g\|_\infty R^2/(2t)$. Let $t\to\infty$ and then $\varepsilon\downarrow0$ to obtain $g(z)=0$, a contradiction. This proves the <fundamental theorem of algebra> without the <maximum modulus principle>.
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