Polynomials with small values in the neighborhoods of zeros in Archimedean and non-Archimedean metrics
https://doi.org/10.22405/2226-8383-2021-22-3-143-153
Abstract
For a positive integer 𝑄 > 0, let 𝐼 ⊂ R denote an interval of length 𝜇1𝐼 = 𝑄−𝛾1 (where 𝜇1 is the Lebesgue measure) and 𝜇2𝐾 = 𝑄−𝛾2 , 𝛾2 > 0 (where 𝜇2 is the Haar measure of a measurable cylinder 𝐾 ⊂ Q𝑝). Let us denote the set of polynomials of degree ≤ 𝑛 and height 𝐻 (𝑃) ≤ 𝑄 as 𝒫𝑛 (𝑄) = {𝑃 ∈ Z[𝑥] : deg 𝑃 ≥ 𝑛, 𝐻 (𝑃) ≤ 𝑄} .
Let 𝒜(𝑛,𝑄) denote the set of real and 𝑝-adic roots of such polynomials 𝑃 (𝑥) lying in the space 𝑉 = 𝐼 ×𝐾. In this paper it is proved that the following inequality holds for a suitable constant
𝑐1 = 𝑐1 (𝑛) and 0 ≤ 𝑣1, 𝑣2 6 1 2 : #𝒜(𝑛,𝑄) > 𝑐1𝑄𝑛+1−𝛾1−𝛾2 .
The proof relies on methods of metric theory of Diophantine approximation developed by V.G. Sprindzuk to prove Mahler’s conjecture and by V.I. Bernik to prove A. Baker’s conjecture.
About the Authors
Artyom Vadimovich Lunevich
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
candidate of physical and mathematical sciences, researcher
Natalya Valentinovna Shamukova
University of Civil Protection of the Ministry of Emergency Situations of Belarus
Belarus
Belarus
candidate of physical and mathematical sciences
References
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Review
For citations:
Lunevich A.V., Shamukova N.V. Polynomials with small values in the neighborhoods of zeros in Archimedean and non-Archimedean metrics. Chebyshevskii Sbornik. 2021;22(3):143-153. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-143-153
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