A_(gcd(m,n))² always divides F_(m,n): multiple roots of the Bremner–Ulas quadrinomial discriminants

Abstract

Bremner and Ulas reduce the divisibility of the quadrinomial xⁿ+xⁿ⁻ᵐ+xⁿ⁻²ᵐ+a by a rational quadratic to the rational points of the hyperelliptic curve s²=F_(m,n)(t), where F_(m,n)=Aₙ₋ₘ²-4Aₙ₋₂ₘAₙ is built from the sequence Aₖ=Aₖ(1,t) ∈ ℤ[t] of their Lemma 3.1, and they conjecture that F_(m,n) has no multiple roots whenever n>2m. We show that this is false and identify the obstruction exactly: A_(gcd(m,n))² divides F_(m,n) for all m,n, over every commutative ring. Since A_(d) is a non-unit precisely for d ≥ 3, the polynomial F_(m,n) fails to be squarefree, and has a multiple complex root, for every pair with n>2m and gcd(m,n) ≥ 3; the smallest is (m,n)=(3,9), where F_(3,9)=(t-1)²(4t³-27t²+18t-3) although 9>2 · 3. We propose the repaired statement gcd(F_(m,n),F_(m,n)')=A_(gcd(m,n)) up to a nonzero constant — equivalently, F_(m,n) is squarefree if and only if gcd(m,n) ≤ 2 — and verify it for all 210 pairs with 3 ≤ n ≤ 30, 0<m, 2m<n by an explicit Bézout certificate per pair, in which both exact divisions are checked rather than assumed and no polynomial gcd algorithm is trusted; an independent computation confirms it for all 1560 pairs with n ≤ 80. Every theorem below is machine-checked in Lean 4; S[sec:verif] says exactly what the machine checks and names the two steps that remain outside it. One caveat is stated in full in S[sub:whichtext]: the published version of the source is closed access, and the refutation is graded against its arXiv e-print.

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