1. CMB 2017 (vol 60 pp. 561)
 Kurdyka, Krzysztof; Paunescu, Laurentiu

Nuij Type Pencils of Hyperbolic Polynomials
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.
Keywords:hyperbolic polynomial, stable polynomial, determinantal representa tion, symmetric Toeplitz matrix Categories:15A15, 30C10, 47A56 

2. CMB 2016 (vol 60 pp. 613)
 Reichstein, Zinovy; Vistoli, Angelo

On the Dimension of the Locus of Determinantal Hypersurfaces
The characteristic polynomial $P_A(x_0, \dots,
x_r)$
of an $r$tuple $A := (A_1, \dots, A_r)$ of $n \times n$matrices
is
defined as
\[ P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r
A_r) \, . \]
We show that if $r \geqslant 3$
and $A := (A_1, \dots, A_r)$ is an $r$tuple of $n \times n$matrices in general position,
then up to conjugacy, there are only finitely many $r$tuples
$A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently,
the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$
is irreducible of dimension $(r1)n^2 + 1$.
Keywords:determinantal hypersurface, matrix invariant, $q$binomial coefficient Categories:14M12, 15A22, 05A10 

3. CMB 2016 (vol 59 pp. 585)
 Lin, Minghua

A Determinantal Inequality Involving Partial Traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes
\mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved
an inequality for Schatten $p$norms:
\[
1+\\mathbf{A}\_p\ge \\tr_1 \mathbf{A}\_p+\\tr_2 \mathbf{A}\_p,
\]
where $\tr_1, \tr_2$ stand for the first and second partial
trace, respectively. As an analogue of his result, we prove a
determinantal inequality
\[
1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.
\]
Keywords:determinantal inequality, partial trace, block matrix Categories:47B65, 15A45, 15A60 

4. CMB 2016 (vol 59 pp. 311)
 Ilten, Nathan; Teitler, Zach

Product Ranks of the $3\times 3$ Determinant and Permanent
We show that the product rank of the $3 \times 3$ determinant
$\det_3$ is $5$,
and the product rank of the $3 \times 3$ permanent
$\operatorname{perm}_3$
is $4$.
As a corollary, we obtain that the tensor rank of $\det_3$ is
$5$ and the tensor rank of $\operatorname{perm}_3$ is $4$.
We show moreover that the border product rank of $\operatorname{perm}_n$ is
larger than $n$ for any $n\geq 3$.
Keywords:product rank, tensor rank, determinant, permanent, Fano schemes Categories:15A21, 15A69, 14M12, 14N15 
