The sharp threshold for the multiplicative Merino-Welsh inequality

It is proved that $T_M(x_*,0)T_M(0,x_*)\ge T_M(1,1)^2$ for every finite matroid $M$ without loops or coloops, and known counterexamples below $x_*$ make the universal threshold sharp. For $2\le x\le x_*$, the stronger bound $$\frac{T_M(x,0)T_M(0,x)}{T_M(1,1)^2}\ge\left(\frac{x^3}{9(x-1)}\right)^{(m-2\kappa)/2}$$ holds, where $m$ is the number of elements and $\kappa$ the number of connected components. Parallel doublings of uniform matroids show the exponential rate is optimal throughout the interval. The proof uses weighted broken circuits, matroid decomposition, and exact checks of rational replacements. The graph conjecture at $x=2$ remains open.

It is proved that $T_M(x_*,0)T_M(0,x_*)\ge T_M(1,1)^2$ for every finite matroid $M$ without loops or coloops, and known counterexamples below $x_*$ make the universal threshold sharp. For $2\le x\le x_*$, the stronger bound $$\frac{T_M(x,0)T_M(0,x)}{T_M(1,1)^2}\ge\left(\frac{x^3}{9(x-1)}\right)^{(m-2\kappa)/2}$$ holds, where $m$ is the number of elements and $\kappa$ the number of connected components. Parallel doublings of uniform matroids show the exponential rate is optimal throughout the interval. The proof uses weighted broken circuits, matroid decomposition, and exact checks of rational replacements. The graph conjecture at $x=2$ remains open.