0524140 - MÚ 2021 RIV GB eng J - Journal Article
Hrubeš, Pavel
On the distribution of runners on a circle.
European Journal of Combinatorics. Roč. 89, October (2020), č. článku 103137. ISSN 0195-6698. E-ISSN 1095-9971
R&D Projects: GA ČR(CZ) GX19-27871X
Institutional support: RVO:67985840
Keywords : computational complexity * distribution of runners on a circle
OECD category: Pure mathematics
Impact factor: 0.847, year: 2020 ; AIS: 0.957, rok: 2020
Method of publishing: Limited access
Result website:
https://doi.org/10.1016/j.ejc.2020.103137DOI: https://doi.org/10.1016/j.ejc.2020.103137

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 bound is asymptotically tight up to a logarithmic factor. The result can be generalized as follows. Let f(x,y) be a complex bivariate polynomial whose Newton polytope has k vertices. Then there exist a∈ℂ∖{0} and a complex sector S={reıθ:r>0,α≤θ≤β} such that the univariate polynomial f(x,a) contains at least [Formula presented]n+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 (2011) implies the conjecture on Newton polytopes of Koiran et al. (2015).
Permanent Link: http://hdl.handle.net/11104/0308495