Polynomial Separation at Roots of Unity

Definitions and mathematical statement

Let $n$ be a positive integer and let $f\in{\mathbb{Z}}[t_1,\ldots,t_n]$. A root of unity is a complex number $z$ such that $z^m=1$ for some positive integer $m$; its order $\operatorname{ord}(z)$ is the least such integer. Write $|\cdot|$ for the usual complex modulus.

The supplied question asks whether there exists a real number $E=E(f)>0$, independent of the order bound and the chosen roots, such that \[ \forall d\in{\mathbb{Z}}_{\ge 1}\quad \forall(z_1,\ldots,z_n)\in{\mathbb{C}}^n,\qquad \left(\operatorname{ord}(z_i)\le d\text{ for every }i\right) \Longrightarrow \left(f(z_1,\ldots,z_n)=0\ \text{or}\ |f(z_1,\ldots,z_n)|>d^{-E}\right), \] where the tuple is restricted to roots of unity.

Unresolved qualification. The displayed quantification over positive integers $d$ makes explicit the natural literal reading of the supplied order bound; the source record does not specify its range. At $d=1$, all $z_i=1$, and the strict inequality requires $|f(1,\ldots,1)|>1$ whenever this value is nonzero. Thus nonzero values of modulus $1$ are excluded by the literal formulation. The record does not resolve whether the intended statement restricts $d$, changes the strict inequality, or includes another qualification. No such repair is adopted here.

Short English statement

The supplied conjecture asks for an exponent depending only on the integer polynomial that bounds every nonzero value at roots of unity away from zero by a power of their order bound. The strict inequality and the unspecified range of $d$, especially the boundary $d=1$, require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30002177$, title Polynomial Separation at Roots of Unity, exact source code catalogue entry. The record supplies proposed year $2012$ and names no proposer.
• Supplied report citation: Low-Dimensional Topology and Number Theory (2013), Oberwolfach Reports, pp. 2541--2596, DOI 10.4171/owr/2012/42; identified in the literature sources as Oberwolfach Report 42, Conjecture 6. https://doi.org/10.4171/owr/2012/42.
• Supplied literature reference: Barber, small sums of five roots of unity. https://doi.org/10.1112/blms.12826.
• Supplied literature reference: Kalogirou, small sums of roots of unity. https://arxiv.org/abs/2607.06098.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that reduction to sums of roots of unity is known, and reports that no general polynomial separation bound has been proved. This is supplied dated curation, not independently verified. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent mathematical verification of this draft.