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 $f=y_n^d$.