I'm not a professional geometer / topologist, so please thanks for your patience :)

# Setup

The following questions are the first in a series of steps I'm undertaking in an attempt to break down a bigger question (MO link) into standalone chewable pieces.

So, let $F$ be a field (e.g $\mathbb R$, $\mathbb C$, $\mathbb F_2$, etc.), $n$, $p$, $q$, and $r$ be positive integers and let $F^{n \times p}$ and $Y \in F^{n \times r}$ be fixed matrices. Consider the system of equations in unknown matrices $U \in F^{p \times q}$ and $V \in F^{q \times r}$ $$ \begin{split} X^T(XUV - Y)V^T &= 0 \in F^{p \times q},\\ U^TX^T(XUV - Y) &= 0 \in F^{q \times r}. \end{split} $$

Note that this is a system of polynomial equations in the entries of the matrices $U$ and $V$. Thus the set of solutions to the above system of equations is an algebraic variety $\Omega$ over the field $F$. I'm interested in the geometry and topology of $\Omega$ (number and nature of isolated points, connected components, etc.).

# Possible questions

- What is a good upper bound on the number of isolated points of $\Omega$ ?
- What is a good upper bound on the average number of isolated points of $\Omega$ when the rows of $X$ and $Y$ are random i.i.d ?

# Observations

- My wild guess is that some kind of [Bézout theorem][2] should be applicable here, but I don't have any experience with that theory.
- The two matricial equations above have a common factor $X^T(XUV-Y)$.
- Symmetries: It's easy to see that if $(U,V) \in \Omega$ and $A$ is an invertible $p$-by-$p$ matrix, then $(UA^{-1},AV) \in \Omega$.
Thus the mapping $(A, (U, V)) \mapsto (UA^{-1},AV)$ defines a group-action on $\Omega$, of $\operatorname{GL}_p$, the
*linear*algebraic group of invertible $p$-by-$p$ matrices. Furthermore this group-action is*regular*. See comment section of this related question (MO link).

Maybe these observations can be exploited somehow in the answer ?

# Updates

- A user has remarked that the group-action above implies that the only possible isolated point of $\Omega$ is the origin $(U=0,V=0)$. Consequently it has also been suggested that a better question is counting the number of isolated
*orbits*. So

# Further questions

*(A)* What's the structure / size of the quotient "$\Omega/\operatorname{GL}_p$" ?

*(B)* More "precisely",

- How many isolated orbits does $\Omega$ have ?
- Can we describe / characterize them ?
- What are their dimensions ?
- etc.

- Same questions as above, but for
*connected*components. - Same questions as above, but for
*irreducible*components - etc.

secondpart of my comment applies: with such a symmetry group, the only possible isolated point is the origin. A more interesting question might thus be to ask about isolatedorbits, but I suppose that depends on your motivation. $\endgroup$ – Joshua Grochow Nov 10 '18 at 0:27