加法的な増大ギャップは対数凹性を保証しない
Additive Augmentation Gaps Do Not Force Log-Concavity
研究概要
独立集合を拡張する規則を一定の加法的余裕だけ緩めても、サイズ別の独立集合数の対数凹性は保証されない。13 元集合上のランク 4 の例は個数列 1, 13, 25, 7, 2 をもち、Xie と Xu のランク依存の予想に反例を与える。さらに、同じ増大パラメータを保ちながら対数凹性の比が 0 に近づく族を構成する。
元の公開問題
Xie–Xu 予想 4.8：緩和された増大条件と対数凹性
Yan-Ting Xie and Shou-Jun Xu: Ultra log-concavity and real-rootedness of dependence polynomials — Conjecture 4.8: rank-dependent log-concavity for independence systems with relaxed augmentation. This is not the classical matroid log-concavity theorem.
原文要旨（英語）
We construct independence systems satisfying a fixed additive relaxation of the matroid augmentation axiom whose independence sequences fail log-concavity. This disproves the rank-dependent extension of Mason's conjecture proposed by Xie and Xu. The construction has two ingredients: the largest gap in a failed augmentation is additive under direct sums, and a tunable low-degree term can survive multiplication by any fixed matroid independence polynomial as a log-concavity obstruction. A thirteen-element example has rank four and independence sequence 1,13,25,7,2. More generally, the augmentation parameter can remain three while the rank tends to infinity. We also construct systems whose rank divided by their ground-set size tends to one, while a consecutive log-concavity ratio tends to zero. All singletons are independent, yet no element can be added to every independent set.
数学の検証
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検証基準