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Let $r > 1$, $\epsilon > 0$, $\eta > 0$ does there always exist a monic polynomial with integer coefficients $P$ such that

  • $P$ has a unique real root $r_0$, s.t $|r_0 - r| < \epsilon$
  • For all other roots of $P$, $r_i$, $|r_i| < \eta$

(I'm expecting this to be false)

Arthur B.
  • 908
  • Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $\epsilon \to 0$ then necessarily $n \to \infty$. Also (with the same argument) for $\eta$ small enough there is no monic integer polynomial whose roots are all $\eta$ close to $0$, so wlog. your polynomial must be irreducible – reuns Jan 25 '19 at 03:22

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