Decide cyclic levels without separable models

Determine which levels of a curve are realized only by cyclic covers that admit no separable model, including cases where the reduction to Weierstrass points fails, by developing a search method beyond the Weierstrass-point reduction used for separable models.

Background

The paper’s main completeness theorem decides precisely the levels admitting a model of the form yn=h(x) with h separable. It does not decide levels realized by cyclic covers whose branch data fails the separability criterion. The paper notes that some such covers can still be recovered when there are two totally ramified points that are Weierstrass points, but this approach fails, for example, for genus three and branch data (1,1,3,3) at level four, where the branch points have the generic vanishing sequence and therefore are not Weierstrass points.

References

Two questions are left open. Levels realised only by cyclic covers that admit no separable model are outside \cref{thm:complete}, and \cref{rem:incomplete} gives branch data, $n=4$ with $(l_j)=(1,1,3,3)$ in genus three, for which the reduction to Weierstrass points fails; deciding those levels needs a different search.

— Deciding superellipticity and computing the Weierstrass normal form  (2609.00672 - Shaska, 1 Sep 2026) in Section Implementation and timings, final paragraph