= Brownian recurrence proof of the fundamental theorem of algebra
If a nonconstant complex polynomial were zero-free, its reciprocal would be an entire bounded function. The real part of its value along planar <Brownian motion> is a bounded martingale, hence converges almost surely. If that real part were nonconstant, continuity would give two disks with separated value ranges, and <recurrence of planar Brownian motion> would force alternate visits to both at arbitrarily large times, contradicting convergence. It is therefore constant; the <Cauchy-Riemann equations> then make the reciprocal constant, a contradiction.
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