  Over the complex numbers, Pl\"ucker's formula computes the number of
inflection points of a linear series of projective dimension $r$ and degree $d$
on a curve of genus $g$. Here we explore the geometric meaning of a natural
analogue of Pl\"ucker's formula in $\mathbb{A}^1$-homotopy theory for certain
linear series on hyperelliptic curves defined over an arbitrary field.
