Complete solution sets and the extremal product for σₙ₋₂(x)=σₙ(x) at n=6 and n=7

Abstract

For integers 1 ≤ x₁ ≤ … ≤ xₙ the equation σₙ₋₂(x)=σₙ(x) between elementary symmetric polynomials is the decomposition of unity Σ_(1 ≤ i<j ≤ n)1/(xᵢxⱼ)=1; write fₙ₋₂(n) for its number of solutions. Kiss, Sándor and Zakarczemny introduced the sequence v₁=1, v₂=2, vₙ₊₁=vₙ²-vₙ+1+v₁… vₙ₋₁, observed that (v₁,…,vₙ₋₁,vₙ-1) solves the equation, listed all solutions for n ≤ 5, and asked whether x₁x₂… xₙ ≤ v₁v₂… vₙ₋₁(vₙ-1) holds for every solution. We answer this affirmatively at the two smallest lengths their paper leaves open. For n=6 and n=7 we determine the solution set completely — f₄(6)=144 and f₅(7)=3598 — and show that the maximum of x₁… xₙ is exactly 1 257 813 360 and 2 881 515 884 053 973 040 respectively, attained at (1,2,4,15,219,47862) and (1,2,4,15,219,47863,2290845186). We also prove, for every n, that the tuple (v₁,…,vₙ₋₁,vₙ-1) satisfies the equation and has product exactly the proposed bound, so that the question is in substance whether that one family is the product-maximiser; and we replace the majorization estimate through which the source obtains finiteness by an elementary prefix inequality, which is what makes the two exhaustive searches terminate. Every theorem below is machine-checked in Lean 4.

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