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Log-concavity in the Newton basis and support of rank-two scattering coefficients

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RT — Representation Theory

Contributed by Qiyue Tang ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We prove that polynomials with nonnegative, interval-supported log-concave Newton coefficients are closed under multiplication and, for integer Newton coefficients, under D(m)↦(D(m)k)D(m)\mapsto\binom{D(m)}k for integers k≥0k\ge0. Coordinatewise Newton multiplication preserves two-variable Schur positivity. We also determine the exact positive Newton support of symmetric rank-two cluster scattering coefficients.

Provenance statement

The research began with a shared Codex session on Conjectures 4, 12 and 13 lasting about 2 hours 58 minutes, followed by eleven ChatGPT rounds on Conjecture 12 with at least 50 hours 42 minutes documented in the surviving records. Codex also reviewed arguments, prepared research prompts and refined the exposition. The main text was reviewed using Rethlas-OSV; the appendices, which list statements without proofs, were excluded from that review. The available proofs, failed approaches and review records are archived at: https://github.com/tqyqiqi1509/c12-newton-research-archive Material packages and source maps: https://github.com/tqyqiqi1509/c12-newton-research-archive/releases/tag/v0.1.0-archive-20261103

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Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentNov 03, 2026