# Counting / characterizing the isolated points of a particular algebraic variety

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 ?

• 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.
• Unless I'm misunderstanding, there seems to be a size mismatch. $X^T X W_0 W_1$ is $p_0 \times p_2$, while $X^T Y$ is $p_0 \times p_1$. Is $Y$ supposed to be $n \times p_2$? – Zach Teitler Nov 9 '18 at 17:39
• Thanks for catching. Yes it was a typo. Fixed. – dohmatob Nov 9 '18 at 17:51
• I don't see why the group action preserves the solution set. E.g. consider the term $X^T Y W_1$ in the first equation... But if you did have such a group action, the only possible isolated point is the origin, since that's the only fixed point of this action. – Joshua Grochow Nov 9 '18 at 21:25
• Oh... If you put $W_1^T$ in the first equation your symmetries work, and then my previous comment applies. – Joshua Grochow Nov 9 '18 at 21:54
• Sorry, I meant the second part 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 isolated orbits, but I suppose that depends on your motivation. – Joshua Grochow Nov 10 '18 at 0:27