Critical-point count for corank-one approximation with one fixed zero

The manuscript proves that for every $n \ge 3$ the variety of singular complex
$n \times n$ matrices with $x_{11} = 0$ has Euclidean distance degree $5n - 7$
for the bilinear squared Frobenius distance, counting critical points on its
smooth locus for generic complex data. After an orthogonal reduction of the
data, the characteristic polynomial of $A(z)A(z)^{\mathsf T}$, where $A(z)$ is
the data matrix with its $(1,1)$ entry replaced by a parameter $z$, is
quadratic in $z$ and defines a smooth plane spectral curve. Critical points
correspond bijectively to the common zeros of this curve and a cubic
stationarity equation, whose resultant in $z$ has degree $5n - 7$. A generic
reducedness argument shows that this degree counts distinct critical points,
and an explicit real family of data shows that the required genericity
conditions hold simultaneously in every dimension.