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How small can polynomials be in an interval of given length? (English) Zbl 1439.11175

In the present article, the main attention is given to two extensions of one lemma from the paper [V. I. Bernik, Acta Arith. 42, 219–253 (1983; Zbl 0482.10049)]. The authors note that this lemma “can be thought of as a quantitative description of the fact that two relatively prime polynomials in \(\mathbb Z[x]\) cannot both have very small absolute values (in terms of their degrees and heights) in an interval unless that interval is extremely short.”
The last-mentioned lemma is improved, an example of its using is given for the case of the number of polynomials with bounded discriminants. Another application is considered.
Auxiliary notations, surveys, and statements are given. Some remarks are noted by the authors.

MSC:

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension

Citations:

Zbl 0482.10049
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Full Text: DOI

References:

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