Integer polynomials and Minkowski’s theorem on linear forms
https://doi.org/10.22405/2226-8383-2022-23-1-45-52
Abstract
In paper Minkowski’s theorem on linear forms [1] is applied to polynomials with integer coefficients
𝑃(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + . . . + 𝑎1𝑥 + 𝑎0 (3)
with degree 𝑑𝑒𝑔𝑃 = 𝑛 and height 𝐻(𝑃) = max06𝑖6𝑛 |𝑎𝑖|. Then, for any 𝑥 ∈ [0, 1) and a natural number 𝑄 > 1, we obtain the inequality
|𝑃(𝑥)| < 𝑐1(𝑛)𝑄−𝑛 (4)
for some 𝑃(𝑥),𝐻(𝑃) ≤ 𝑄. Inequality (4) means that the entire interval [0, 1) can be covered by intervals 𝐼𝑖, 𝑖 = 1, 2, . . . at all points of which inequality (4) is true. An answer is given to the question about the size of the 𝐼𝑖 intervals. The main result of this paper is proof of the following statement.
For any 𝑣, 0 ≤ 𝑣 < (𝑛+1)/3 , there is an interval 𝐽𝑘, 𝑘 = 1, . . . ,𝐾, such that for all 𝑥 ∈ 𝐽𝑘, the inequality (4) holds and, moreover,
𝑐2𝑄−𝑛−1+𝑣 < 𝜇𝐽𝑘 < 𝑐3𝑄−𝑛−1+𝑣.
About the Authors
Vasilii Ivanovich Bernik
Institute of Mathematics NAS Belarus
Belarus
Belarus
doctor of physical and mathematical sciences, professor
Irina Alexandrovna Korlyukova
Grodno State University
Belarus
Belarus
candidate of physical and mathematical sciences, associate professor
Alexey Sergeevich Kudin
Institute of Mathematics NAS Belarus
Belarus
Belarus
candidate of physical and mathematical sciences
Anastasia Vladimirovna Titova
Institute of Mathematics NAS Belarus
Russian Federation
Russian Federation
postgraduate student
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Review
For citations:
Bernik V.I., Korlyukova I.A., Kudin A.S., Titova A.V. Integer polynomials and Minkowski’s theorem on linear forms. Chebyshevskii Sbornik. 2022;23(1):45-52. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-1-45-52
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