Nonnegative Cyclotomic Products and Square-Sided Dice
Overview
Cyclotomic polynomials encode primitive roots of unity. The paper classifies the nonnegative products in a family determined by powers of two distinct primes, resolving Fiore–Nasr–Stone Conjecture 5.7 on relabeling square-sided dice. It also proves a sharp multiplier degree bound and all equality cases. The stated mathematical results have public Lean 4 proofs over integer polynomials.
Original public questions
Fiore–Nasr–Stone Conjecture 5.7: square-sided dice
Evelyn Fiore, George D. Nasr and Cooper Stone: Dice Relabeling Using Square-Sided Dice — Conjecture 5.7: coefficient positivity of the cyclotomic product Q_01 in square-sided dice relabeling.
Original abstract (English)
We determine when a family of products of cyclotomic polynomials has nonnegative coefficients. For distinct primes p, q and positive integers a, b, the product Phi_(q^b)(x)^2 Phi_(p^a q^b)(x) is nonnegative coefficientwise precisely when a = 1, or when a = 2, p = 2 and q is congruent to 1 modulo 4. The case a = 2, b = 1 proves a conjecture of Fiore, Nasr and Stone arising from relabeling dice with square numbers of sides. The proof uses the classical decomposition of nonnegative vanishing sums of roots of unity whose orders have two prime divisors. For products with nonnegative coefficients, this gives a sharp degree bound on integer multipliers whose coefficient sums are not divisible by a prescribed prime, together with all equality cases. At the exceptional boundary, a positive factorization gives the admissible dice polynomial directly. We also obtain a uniform obstruction for higher powers of the first cyclotomic factor.
Mathematical review
The principal conclusions have a public kernel-checked proof package and reviewed correspondence with the paper. This is distinct from external peer review.
Review standard