发表机构
School of Mathematical Sciences Xiamen University（厦门大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过直接归纳证明，解决了Csikvári关于Merino--Welsh型不等式的猜想，确定了常数$c_*$等于$x_0$，即多项式$x^3-9(x-1)$的最大实根。
AI 中文摘要
受Merino--Welsh猜想的启发，我们考虑最小的$c\ge0$，记为$c_*$，使得对于每个无环且无余环的有限拟阵$M$，不等式$T_M(c,0)T_M(0,c)\ge T_M(1,1)^2$成立。Beke、Csáji、Csikvári和Pituk [\emph{Adv. Math.} \textbf{446} (2024), 109674]构造的反例给出了下界$x_0$，其中$x_0\approx2.22668$是多项式$x^3-9(x-1)$的最大实根。后来，Csikvári [\emph{European J. Combin.} \textbf{137} (2026), 104402]将这个常数的已知上界改进到$2.35$，并猜想上述不等式在$c=x_0$处成立。我们给出了这个猜想的直接归纳证明，从而表明$c_*=x_0$。
英文摘要
Motivated by the Merino--Welsh conjecture, we consider the smallest $c\ge0$, denoted by $c_*$, for which the inequality $T_M(c,0)T_M(0,c)\ge T_M(1,1)^2$ holds for every loopless and coloopless finite matroid $M$. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [\emph{Adv. Math.} \textbf{446} (2024), 109674] give the lower bound $x_0$, where $x_0\approx2.22668$ is the largest real root of the polynomial $x^3-9(x-1)$. Later, Csikvári [\emph{European J. Combin.} \textbf{137} (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at $c=x_0$. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that $c_*=x_0$, without computer-assisted finite verification.
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