The locus of polynomials with specified root multiplicities Let $\mathcal{P}_d\cong\mathbb{A}^d$ denote the set of monic degree $d$ polynomials defined over an algebraically closed field of characteristic $0$, where we identify $f(x)$ with its coefficients.  The multiplicities of the roots of $f(x)\in\mathcal{P}_d$ defines a partition $\pi(f)$ of $d$. For example, if $f(x)=(x-\alpha)^d$, then $\pi(f)=(d)$, and if $f(x)$ has distinct roots, then $\pi(f)=(1^d)$.
For any partition $\sigma$ of $d$, the set
$$  \mathcal{P}_d(\sigma) := \bigl\{ f\in\mathcal{P}_d : \pi(f)=\sigma\bigr\} $$
is a quasiprojective subvariety of $\mathbb{A}^d$. (This follows from elimination theory.) For example, $\mathcal{P}_d(d)$ is a curve, while $\mathcal{P}_d(1^{d-2},2)$ is an open subset of the discriminant locus $\bigl\{f\in\mathcal{P}_d:\operatorname{Disc}(f)=0\bigr\}$.


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*Do these varieties $\mathcal{P}_d(\sigma)$ have a name? My best guess was discriminantal variety, but that term does not seem to be in use.

*Where have these varieties been studied? Specific references would be appreciated.
 A: I may be wrong but I think "coincident root loci" mentioned in Gjergjji's comment was coined by my coauthor Jaydeep Chipalkatti. The ideals of such varieties are in general poorly understood. 
For a general study see:


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*J. Chipalkatti, "On equations defining Coincident Root loci", J. Algebra 267 (2003), no. 1, 246-271.

*J. Chipalkatti, "Invariant equations defining coincident root loci", Archiv der Math. 83 (2004), no. 5, 422-428. 

*H. Lee and B. Sturmfels, "Duality of multiple root loci", J. Algebra 446 (2016), 499-526.


For particular cases, the following might be also of interest.


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*A. Abdesselam and J. Chipalkatti, "Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture", Adv. Math. 208 (2007), no. 2, 491-520. It has a description of the ideal for a partition with two equal parts.

*A. Abdesselam and J. Chipalkatti, "The bipartite Brill-Gordan locus and angular momentum", Transformation Groups 11 (2006),
no. 3, 341-370. It has a description of the ideal for a partition with two unequal parts.

*A. Abdesselam and J. Chipalkatti, "On Hilbert Covariants"
Canadian J. Math. 66 (2014), no. 1, 3-30. It has several set-theoretic systems of equations for the varieties corresponding to rectangular partitions, and a conjecture about minimal degree of generators for the ideals. This is an $SL_2$ analogue/toy version of the Foulkes-Howe conjecture.


There is also work about these varieties from a topological point of view, e.g.,
F. Napolitano,
"On some topological invariants of algebraic functions associated to the Young stratification of polynomials"
Topology Appl. 134 (2003), no. 3, 189-201. 
The relation to Hurwitz stacks is studied in: J. Bertin and M. Romagny,
"Champs de Hurwitz", Mémoires SMF, no. 125-126 (2011), 219 p. See also here for the arXiv version. The relevant result is Theorem 9.16 relating coincident root loci and Hurwitz stacks of cyclic coverings of $\mathbb{P}^1$.
Also, an interesting geometric approach to these varieties is in: G. Katz "How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties", Expositiones Math. 21 (2003),
no. 3, 219-261.
Of course, this is by no means an exhaustive bibliography. There are many references I left out in this short MO post.  
