The closure of the nonreal roots of bounded-height polynomials is connected exactly at pronic heights
Abstract
Let C_h be the closure of the nonreal roots of integer polynomials with coefficients of modulus at most h. With the nonzero real roots included, the closure is connected at every height; without them, connectedness becomes arithmetic. We prove that C_h is connected exactly when h=q(q-1) is a pronic number 2, 6, 12, 20, ..., and has at least five components otherwise. At height two this confirms a twenty-year-old numerical observation of Shmerkin and Solomyak on double zeros of power series: apart from 1/2 they form the interval $[1/Ï,1)=C_2\cap(0,1)$, with $Ï$ the tribonacci constant. Disconnection is almost always infinite. For (p-1)^2<h<=p^2-3, $h\neq p(p-1)$, infinitely many components with nonempty interior accumulate at $\pm p$ and $\pm 1/p$, and when h<p(p-1) these four points are themselves components. Next to each square q^2>=16, at least two of the three heights q^2-2, q^2-1, q^2 have infinitely many components accumulating at a common point of (1/(q+1),1/q). Explicit series give infinitely many components at every nonpronic height below 167 other than 1, 3, 4, 9, 25, 63 and 81, and in general at most sqrt(X)+1 nonpronic heights h<=X can have finitely many components, although C_h is a Peano compactum at every height. The proofs compare values and derivatives of all admissible power series near a real double zero; the comparison sets are open and closed in the whole core, so components separated near the axis cannot reconnect elsewhere. On the real axis, the double zeros at heights 1, 3, 4, 7 form sets with infinitely many components; at height one this proves a conjecture of Shmerkin and Solomyak and one of Solomyak on the connectedness locus for pairs of affine maps.引用本文复制引用
Bernat Espigule.The closure of the nonreal roots of bounded-height polynomials is connected exactly at pronic heights[EB/OL].(2026-11-10)[2026-11-12].https://arxiv.org/abs/2611.14517.学科分类
数学
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