Efficient computation of maximal orders in Artin–Schreier extensions

We describe an algorithm, linear in the degree of the field, for computing pseudo bases for integral closures of holomorphy rings in Artin–Schreier extensions of global function fields and a similar algorithm, also linear in the degree of the field, for computing pseudo bases for S-maximal orders of Artin–Schreier extensions of global function fields. We give examples comparing the running time of our algorithm to that of Round 2 and Fraatz (2005).