The following question is motivated by the study of a stability border for a robust linear time-invariant control system.

Let us we have an affine family of $n\times n$ matrices with indeterminate entries.

$$ A=A_0+p_1A_1+\ldots+p_kA_k,\qquad A_i=\begin{pmatrix}a_{i,11}\ldots a_{i,1n}\\\ldots\\a_{i,n1}\ldots a_{i,nn}\end{pmatrix} $$

Let $\chi_A(\lambda)$ be a characteristic polynomial of $A$. Let us define $R= \Re( \chi_A(i\sqrt{\omega})), I=\frac{\Im(\chi_A(i\sqrt{\omega}))}{\sqrt{\omega}}.$ Both $R$ and $I$ are a polynomials depending on $\omega.$ Let $S$ be a resultant of $R$ and $I$ as $\omega$-polynomials.

Is $S$ irreducible as a polynomial from $(k+1)(n^2+1)$ variables $a_{i,jk}, p_i$? Is it true that if matrices $A_i$ are, in some sense, in "general position" then $S$ is irreducible as a polynomial in $p_i$?

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.

Let us we have an affine family of $n\times n$ matrices with indeterminate ($\mathbb{R}$-valued) entries.

$$ A=A_0+p_1A_1+\ldots+p_kA_k,\qquad A_i=\begin{pmatrix}a_{i,11}\ldots a_{i,1n}\\\ldots\\a_{i,n1}\ldots a_{i,nn}\end{pmatrix} $$

Let $\chi_A(\lambda)$ be a characteristic polynomial of $A$. Let us define $R= \Re( \chi_A(i\sqrt{\omega})), I=\frac{\Im(\chi_A(i\sqrt{\omega}))}{\sqrt{\omega}}.$ Both $R$ and $I$ are a polynomials depending on $\omega.$ Let $S$ be a resultant of $R$ and $I$ as $\omega$-polynomials.

Is $S$ irreducible as a polynomial from $(k+1)(n^2+1)$ variables $a_{i,jk}, p_i$? Is it true that if matrices $A_i$ are, in some sense, in "general position" then $S$ is irreducible as a polynomial in $p_i$?

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.

Let us we have an affine family of $n\times n$ matrices with indeterminate entries.

$$ A=A_0+p_1A_1+\ldots+p_kA_k,\qquad A_i=\begin{pmatrix}a_{i,11}\ldots a_{i,1n}\\\ldots\\a_{i,n1}\ldots a_{i,nn}\end{pmatrix} $$

Let $\chi_A(\lambda)$ be a characteristic polynomial of $A$. Let us define $R= \Re( \chi_A(i\sqrt{\omega})), I=\frac{\Im(\chi_A(i\sqrt{\omega}))}{\sqrt{\omega}}.$ Both $R$ and $I$ are a polynomials depending on $\omega.$ Let $S$ be a resultant of $R$ and $I$ as an $\omega$-polynomials.

Is $S$ irreducible as a polynomial from $(k+1)(n^2+1)$ variables $a_{i,jk}, p_i$? Is it true that if matrices $A_i$ is, in some sense, in "general position" then $S$ is irreducible as a polynomial in $p_i$?

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.

Let us we have an affine family of $n\times n$ matrices with indeterminate entries.

$$ A=A_0+p_1A_1+\ldots+p_kA_k,\qquad A_i=\begin{pmatrix}a_{i,11}\ldots a_{i,1n}\\\ldots\\a_{i,n1}\ldots a_{i,nn}\end{pmatrix} $$

Let $\chi_A(\lambda)$ be a characteristic polynomial of $A$. Let us define $R= \Re( \chi_A(i\sqrt{\omega})), I=\frac{\Im(\chi_A(i\sqrt{\omega}))}{\sqrt{\omega}}.$ Both $R$ and $I$ are a polynomials depending on $\omega.$ Let $S$ be a resultant of $R$ and $I$ as $\omega$-polynomials.

Is $S$ irreducible as a polynomial from $(k+1)(n^2+1)$ variables $a_{i,jk}, p_i$? Is it true that if matrices $A_i$ are, in some sense, in "general position" then $S$ is irreducible as a polynomial in $p_i$?