Subalgebras of Matrix Algebras Generated by Companion Matrices

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given

A Closed Formula for the Product in Simple Integral Extensions

Let $\xi$ be an algebraic number and let $\alpha,\beta\in \mathbb Q[\xi]$. An explicit formula for the coordinates of the product $\alpha\beta$ is given in terms of the coordinates of $\alpha$ and $\beta$ and the companion matrix of the minimal polynomial of $\xi$. The formula as well as its proof extend to fairly general simple integral extensions

Structural Complexity and Phonon Physics in 2D Arsenenes

In the quest for stable 2D arsenic phases, four different structures have been recently claimed to be stable. We show that, due to phonon contributions, the relative stability of those structures differs from previous reports and depends crucially on temperature. We also show that one of those four phases is in fact mechanically unstable. Furthermore, our results challenge the common assumption of an inverse correlation between structural complexity and thermal conductivity. Instead, a richer picture emerges from our results, showing how harmonic interactions, anharmonicity and symmetries all play a role in modulating thermal conduction in arsenenes. More generally, our conclusions highlight how vibrational properties are an essential element to be carefully taken into account in theoretical searches for new 2D materials