A Characterization of Quasi-homogeneous Bivariate Polynomials

If a reduced bivariate polynomial is quasi-homogeneous, then its discriminant is a monomial. Over fields of characteristic $0$, we show that if one adds another simple condition, this becomes an equivalence. We also give a third equivalent condition that is stated geometrically.
Comment: Fixed minor mistakes, reordered the introduction, streamlined some terminology and extended the result to large positive characteristic