Polynomials with small values in the neighborhoods of zeros in Archimedean and non-Archimedean metrics

A. V. Lunevich ; N. V. Shamukova

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 143-153

Résumé

For a positive integer $Q>0$, let $I\subset \mathbb{R}$ denote an interval of length $\mu_1 I=Q^{-\gamma_1}$ (where $\mu_1$ is the Lebesgue measure) and $\mu_2 K=Q^{-\gamma_2}, \ \gamma_2>0$ (where $\mu_2$ is the Haar measure of a measurable cylinder $K \subset \mathbb{Q}_p$). Let us denote the set of polynomials of degree $\leq n$ and height $H\left(P\right)\leq Q$ as $$ \mathcal{P}_n\left(Q\right)=\left\{P\in \mathbb{Z}[x]\ :\ \deg{P}\geq n,\ H\left(P\right)\leq Q\right\}. $$ Let $\mathcal{A}\left(n,Q\right)$ denote the set of real and $p$-adic roots of such polynomials $P\left(x\right)$ lying in the space $V=I\times K$. In this paper it is proved that the following inequality holds for a suitable constant $c_1=c_1\left(n\right)$ and $0\leq v_1, v_2\le \frac{1}{2}$: $$ \#\mathcal{A}\left(n,Q\right)\ge c_1 Q^{n+1-\gamma_1-\gamma_2}. $$ 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.

Détail

Keywords: Lebesgue measure, Haar measure, algebraic numbers, Diophantine approximation, irreducible polynomials.