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In this paper, we explore the inflectionary behavior of linear series on superelliptic curves $X$ over fields of arbitrary characteristic. Here we give a precise description of the inflection of linear series over the ramification locus of the superelliptic projection; and we initiate a study of those inflectionary varieties that parameterize the inflection points of linear series on $X$ supported away from the superelliptic ramification locus that is predicated on the behavior of their Newton polytopes., Comment: 34 pages, 6 figures

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., Comment: Final, enhanced version, to appear in Math. Nachrichten; 30 pages, 4 figures