On the distribution of runners on a circle
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Consider n runners running on a circular track of unit length with constant speeds such that k of the speeds are distinct. We show that, at some time, there will exist a sector S which contains at least |S|n+ Ω(√(k)) runners. The result can be generalized as follows. Let f(x,y) be a complex bivariate polynomial whose Newton polytope has k vertices. Then there exists a∈C∖{0} and a complex sector S={re^θ: r>0, α≤θ≤β} such that the univariate polynomial f(x,a) contains at least β-α/2πn+Ω(√(k)) non-zero roots in S (where n is the total number of such roots and 0≤ (β-α)≤ 2π). This shows that the Real τ-Conjecture of Koiran implies the conjecture on Newton polytopes of Koiran et al.
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