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On the number of polynomials with small discriminants in the Euclidean and \(p\)-adic metrics. (English) Zbl 1275.11109
The main result of this paper is a lower bound for the number of polynomials with integer coefficients of bounded degree and height satisfying the property that their discriminant is ‘small’ and divisible by a large power of a fixed prime. The precise statement uses the following notation. Let \(P_n(Q)\) denote the set of non-zero polynomials with integer coefficients of degree \(\leq n\) and height \(\leq Q\). The height of a polynomial is defined as the maximum of the absolute values of its coefficients. Let \(D(P)\) denote the discriminant of a polynomial \(P\). For \(v\geq0\) let \[ \mathcal{P}_n(Q,v)=\{P\in P_n(Q):\;1\leq|D(P)|< Q^{2n-2-2v},\;|D(P)|_p<Q^{-2v}\}, \] where \(|\cdot|_p\) denotes the \(p\)-adic valuation. The authors prove that for \(n\geq3\) and \(0\leq v<1/3\) and all sufficiently large \(Q\) one has that \[ \#\mathcal{P}_n(Q,v)\geq C\,Q^{n+1-4v}. \]

MSC:
11J83 Metric theory
11J54 Small fractional parts of polynomials and generalizations
11J61 Approximation in non-Archimedean valuations
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