\dots, v_{p-n_2})$.

1.  \[equal\] If ${\mathbf c}={\mathbf d}$, ${\mathbf u}={\mathbf v}$, then there exists a pencil $P(s)\in {\mathbb F}[s]^{p \times q}$ of $\rank (P(s))=1$ such that $A(s)+P(s){\ensuremath{\stackrel{s.e.}{\sim}}}B(s)$ if and only if $$\label{eqintfihr1}
    \psi_{i-1}(s, t)\mid\phi_i(s, t)\mid\psi_{i+1}(s, t), \quad 1\leq i \leq n.$$

2.  \[rowequal\] If ${\mathbf c}\neq {\mathbf d}$, ${\mathbf u}={\mathbf v}$, let $$\ell=\max\{i \; :\; c_i\neq d_i\},$$ $$f=\max\{i\in\{1, \dots, \ell\}\; : \; c_i<d_{i-1}\} \quad (d_0=+\infty),$$ $$f'=\max\{i\in\{1, \dots, \ell\}\; : \; d_i<c_{i-1}\} \quad (c_0=+\infty),$$ $$G=n-1-\sum_{i=1}^{n-1}\deg(\gcd(\phi_{i+1}(s, t),\psi_{i+1}(s, t)))- \sum_{i=1}^{p-n}u_i,$$ $$T=n-\sum_{i=1}^{n}\deg(\lcm(\phi_{i}(s, t),\psi_{i}(s, t)))- \sum_{i=1}^{p-n}u_i.$$ Then there exists a pencil $P(s)\in {\mathbb F}[s]^{p \times q}$ of $\rank (P(s))=1$ such that $A(s)+P(s){\ensuremath{\stackrel{s.e.}{\sim}}}B(s)$ if and only if (\[eqintfihr1\]) and one of the two following conditions holds: $$\label{eqGg}
     G\leq \sum_{i=1}^{q-n}\min\{c_i, d_i\}+ \max\{c_f, d_{f'}\},$$ or $$\label{eqT}
    T\geq \sum_{i=1}^{q-n}\max\{c_i, d_i\}- \max\{c_f, d_{f'}\}.$$

3.  \[colequalcdrns\] If ${\mathbf c}= {\mathbf d}$, ${\mathbf u}\neq {\mathbf v}$, let $$\bar \ell=\max\{i \; :\; u_i\neq v_i\},$$ $$\bar f=\max\{i\in\{1, \dots, \bar \ell\}\; : \; u_i<v_{i-1}\} \quad (v_0=+\infty),$$ $$\bar f'=\max\{i\in\{1, \dots, \bar \ell\}\; : \; v_i<u_{i-1}\} \quad (u_0=+\infty),$$ $$\bar G=:n-1-\sum_{i=1}^{n-1}\deg(\gcd(\phi_{i+1}(s, t),\psi_{i+1}(s, t)))-\sum_{i=1}^{q-n}c_i,$$ $$\bar T=n-\sum_{i=1}^{n}\deg(\lcm(\phi_{i}(s, t),\psi_{i}(s, t)))-\sum_{i=1}^{q-n}c_i.$$ Then there exists a pencil $P(s)\in {\mathbb F}[s]^{p \times q}$ of $\rank (P(s))=1$ such that $A(s)+P(s){\ensuremath{\stackrel{s.e.}{\sim}}}B(s)$ if and only if (\[eqintfihr1\]) and one of the two following conditions holds: $$\label{eqbarGg}
     \bar 