An arithmetic count of osculating lines

We say that a line in ${\mathbb{P}}_k^{n+1}$ is osculating to a hypersurface $Y$ if they meet with contact order $n+1$. When $k=\mathbb{C}$, it is known that through a fixed point of $Y$, there are exactly $n!$ of such lines. Under some parity condition on $n$ and $\mathrm{d}\mathrm{e}\mathrm{g}(Y)$, we define a quadratically enriched count of these lines over any perfect field $k$. The count takes values in the Grothendieck–Witt ring of quadratic forms over $k$ and depends linearly on $\mathrm{d}\mathrm{e}\mathrm{g}(Y)$.