is formed by a chain of homogeneous polynomials $\Gamma_1(s,t) \mid \dots \mid \Gamma_n(s,t),\ \Gamma_i(s,t) \in {\mathbb F}[s,t], \ 1\leq i\leq n$, called the [*homogeneous invariant factors*]{}, and two collections of nonnegative integers $c_1\geq \dots \geq c_{q-n}$ and $u_1\geq \dots \geq u_{p-n}$, called the [*column and row minimal indices*]{} of the pencil, respectively. In turn, the homogeneous invariant factors are determined by a chain of polynomials $\gamma_1(s)\mid \ldots \mid \gamma_n(s)$ in ${\mathbb F}[s]$, called the invariant factors, and a chain of polynomials $t^{k_1}\mid \ldots \mid t^{k_n}$ in ${\mathbb F}[t]$, called the infinite elementary divisors. In fact, we can write $$\Gamma_i(s,t)=t^{k_i}t^{\deg(\gamma_i(s))}\gamma_i\left(\frac st\right), \ 1\leq i\leq n.$$ The associated canonical form is the Kronecker canonical form. For details see [@Friedland80 Ch. 2] or [@Ga74 Ch. 12] for infinite fields, and [@Ro03 Ch. 2] for arbitrary fields. In what follows we will work with the homogeneous invariant factors. We will take $\Gamma_i(s,t)=1$ ($\gamma_i(s)=1$) whenever $i<1$ and $\Gamma_i(s,t)=0$ ($\gamma_i(s)=0$) when $i>n$. The sum of the degrees of the homogeneous invariant factors plus the sum of the minimal indices is equal to the rank of the pencil. Also, if $T(s)=G(s)^T$, then $G(s)$ and $T(s)$ share the homogeneous invariant factors and have interchanged minimal indices, i.e., the 