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Let $mathcalP_dcongmathbbA^d$ 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 $f(x)inmathcalP_d$ 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
$$ mathcalP_d(sigma) := bigl finmathcalP_d : pi(f)=sigmabigr $$
is a quasiprojective subvariety of $mathbbA^d$. (This follows from elimination theory.) For example, $mathcalP_d(d)$ is a curve, while $mathcalP_d(1^d-2,2)$ is an open subset of the discriminant locus $biglfinmathcalP_d:operatornameDisc(f)=0bigr$.

• Do these varieties $mathcalP_d(sigma)$ 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.

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:

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


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


3. 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.

1. 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.


2. 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.


3. A. Abdesselam and J. Chipalkatti, "On Hilbert Covariants"
   Canadian J. Math. 66 (2014), no. 1, 3-30. It has several set-theoretic sets of equations for the variety with rectangular partition, and a conjecture about minimal degree of generators for the ideal. 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.

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.