Tan–Zhang: nonnegative cyclotomic multiples

---

Held copy and library card.

Hu Tan and Ying Zhang, "Sharp diameter bounds for nonnegative cyclotomic multiples," arXiv:2609.07646 (2026).

Tan and Zhang prove sharp support-diameter bounds for polynomials with nonnegative real coefficients divisible by a cyclotomic polynomial (Theorem 1.1: $D(F)\ge(p-1)N/p$, $p$ the least prime divisor of $N$, with equality exactly for $cX^b(1+X^{N/p}+\cdots+X^{(p-1)N/p})$). Evaluated at a primitive $N$th root of unity, such a polynomial gives a vanishing sum of $N$th roots of unity with nonnegative weights, its exponent support recording where the mass lies. The extremal constructions and sharp bounds provide another way to constrain the geometry of positive relations.