# The Arc Length of the Lemniscate { | p(z) | = 1 }

Borwein, Peter (1993) The Arc Length of the Lemniscate { | p(z) | = 1 }. [Preprint]

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We show that the length of the set $$\{z \in {\Bbb C}: \left|\Pi^n_{i=1} (z - \alpha_i) \right| = 1\}$$ is at most $8\pi e n$. This gives the correct rate of growth in a long standing open problem of Erd\"os, Herzog and Piranian and improves the previous bound of $74n^2$ due to Pommerenke.