Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct consequence of Borsuk-Ulam theorem, this leds to inequality:
$$ \binom{n}{d}-n\geq2(d-1)(n-d-1) $$
In order to apply Morse-Sard theorem (and deduce that the map I've built is not surjective), I'm interested in the proof of the strict inequality. Does someone help me? Here's the complete statement:
Let $n\in\mathbb{N},$ $n\geq5$ and let $d\in\mathbb{N},$ $1<d<n-1.$ Then
$$ \binom{n}{d}-n>2(d-1)(n-d-1) $$
$\textit{Note:}$ if you think also the (weak) inequality does not hold, please let me know. Perhaps there is a bad mistake in my work.