TITLE: Less discriminating discriminants
QUESTION [7 upvotes]: 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...

REPLY [4 votes]: For partitions of the special form $\lambda=(k,1,1,\dots,1)$ (i.e., hook shapes) there is a very explicit description of the ideal of definition of the corresponding set of polynomials (i.e., those polynomials with a root of order $\geq k$).  Namely, it is the span of the set of (symmetric) Jack polynomials  $f_\mu$, with parameter specialized to $-1/k$, and where $\mu$ ranges over all partitions satisfying
$$\mu_i-\mu_{i+k-1} \geq 2$$ for all $i$.  This is the main result of the paper http://arxiv.org/pdf/math/0112127 by Feigin, Jimbo, Miwa, and Mukhin.
For the application to degree, one might note that this computes the Hilbert function of this ideal quite explicitly in combinatorial terms, though for your problem you are probably interested in the grading where each elementary symmetric function has degree one, and I don't know, off the top of my head, how to extract that information.  Maybe I'll post a separate question so we can see if anywhere else here knows.