# Monotonicity of a ratio of complete homogeneous symmetric polynomials

Open problem settled, formally verified in Lean 4. Vamshi Jandhyala.

> Ait-Haddou's ratio of complete homogeneous symmetric polynomials increases, for every degree and position.

Canonical: https://vamshij.com/research/symmetric-polynomial-ratio
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/symmetric-polynomial-ratio
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/symmetric-polynomial-ratio/paper/paper.pdf

Let `H_i(x) = h_ℓ(1, …, 1, x, …, x)` be the complete homogeneous symmetric polynomial of degree `ℓ` in `n` variables, with `i` of them equal to `x` and the rest equal to `1`. Rachid Ait-Haddou conjectured on MathOverflow (question 467261, 2024) that `Ψ_{i,n} = H_i² / (H_{i+1} H_{i-1})` is strictly increasing on `(1, ∞)` for `2 ≤ i ≤ n - 2`. The note proves it for every `ℓ ≥ 1` and every `1 ≤ i ≤ n - 1`. Treating `i` as a real variable `t`, the logarithmic derivative of `Ψ_{i,n}` is a positive multiple of a second difference in `t` of `P_{ℓ-1}/P_ℓ`, where the `P_ℓ` form a Meixner family; that ratio is convex because the zeros interlace.

Preprint v1, 9 October 2026, not peer reviewed. Theorem 1 and the lemmas of its proof are also formally verified in Lean 4 (`lean/`, standard axioms only). The exposition has not yet been independently reviewed. The author used an AI tool (Claude, Anthropic) in this work, as described in the paper's acknowledgements, and is responsible for its content.