On the distribution of runners on a circle

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00524140" target="_blank" >RIV/67985840:_____/20:00524140 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.ejc.2020.103137" target="_blank" >https://doi.org/10.1016/j.ejc.2020.103137</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.ejc.2020.103137" target="_blank" >10.1016/j.ejc.2020.103137</a>

Result language
angličtina
Original language name
On the distribution of runners on a circle
Original language description
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).
Czech name
—
Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GX19-27871X" target="_blank" >GX19-27871X: Efficient approximation algorithms and circuit complexity</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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