One-element differential standard bases with respect to inverse lexicographical orderings

We give a simplified proof of the following fact: for all nonnegative integers n and d the monomial y n d forms a differential standard basis of the ideal [ y n d ]. In contrast to Levi’s combinatorial proof, in this proof we use the Gröbner bases technique. Under some assumptions we prove the converse result: if an isobaric polynomial f forms a differential standard basis of [ f ], then f = y n d .