Suppose that I want to know whether a polynomial $P(z)$ has a root with multiplicity at least three. This is obviously an algebraic condition, but is there some reasonably concise set of conditions defining the variety (in the space of coefficients)? This must have been studied by the ancients. It is clearly necessary that the discriminant vanish, and also that the resultant of the polynomial $P$ and the second derivative $P^{\prime\prime}$ vanish, but, just as obviously, not sufficient...

**EDIT** Abhinav certainly gives a nice answer to the question, but the question I would REALLY like to know the answer to is: what is the degree of the variety as a function of the partition (as in @Gjergji's answer). Maybe I should read the reference...