 Brownian proof of the fundamental theorem of algebra

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 $\mathbb{P}(|z+B_t|\le R)$ by $R^2/(2t)$. Outside a sufficiently large disk, $|g|\le \epsilon$. Hence $|g(z)|\le \epsilon +\Vert g{\Vert }_{\infty }{R}^2/(2t)$. Let $t\to \infty$ and then $\epsilon \downarrow 0$ to obtain $g(z)=0$, a contradiction. This proves the fundamental theorem of algebra without the maximum modulus principle.