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We prove a version of Clifford’s theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens’ theorem for metric graphs.
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