G\leq \sum_{i=1}^{p-n}\min\{u_i, v_i\}+ \max\{u_{\bar f}, v_{\bar f'}\},$$ or $$\label{eqbarT}
    \bar T\geq \sum_{i=1}^{p-n}\max\{u_i, v_i\}- \max\{u_{\bar f}, v_{\bar f'}\}.$$

4.  \[inequal\] If ${\mathbf c}\neq {\mathbf d}$, ${\mathbf u}\neq {\mathbf v}$, 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 there exist homogeneous polynomials $\pi_1^1(s,t)\mid \dots  \mid \pi_{n}^1(s,t)$ such that $$\label{boca}
    \lcm(\phi_i(s,t), \psi_i(s,t))\mid \pi_i^1(s,t)\mid \gcd(\phi_{i+1}(s,t), \psi_{i+1}(s,t)), \quad 1\leq i \leq n.$$ and one of the four following conditions holds:

    1.  $$\label{cdsr}
        {\mathbf c}\prec'{\mathbf d}, \quad {\mathbf u}\prec'{\mathbf v},$$

        $$\label{sumpi1}
        \sum_{i=1}^{n} \deg(\pi_i^1(s,t))=n-\sum_{i=1}^{q-n_1}c_i-\sum_{i=1}^{p-n_2}v_i.$$

    2.  $$\label{dcrs}
        {\mathbf d}\prec'{\mathbf c}, \quad {\mathbf v}\prec'{\mathbf u},$$

        $$\label{sumpi1t}
        \sum_{i=1}^{n} \deg(\pi_i^1(s,t))=n-\sum_{i=1}^{q-n_2}d_i-\sum_{i=1}^{p-n_1}u_i.$$

    3.  (\[cdsr\]) and (\[sumpi1t\]).

    4.  (\[dcrs\]) and (\[sumpi1\]).

[*Proof.*]{} . Let us assume that there exists a pencil $P(s)\in {\mathbb F}[s]^{p\times q}$ of $\rank P(s)=1$ such that $A(s)+P(s){\ensuremath{\stackrel{s.e.}{\sim}}}B(s)$. By Lemma \[lemmaeq\], one of the two following conditions holds:

1.  There exist pencils $a(s), b(s)\in{\mathbb F}[s]^{1\times q}$ and $A_{21}(s)\in {\mathbb F}^{(p-1)\times q}$ such that $ A(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}a(s)\\A_{21}(s)\end{bmatrix}$ and $B(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}b(s)\\A_{21}(s)\end{bmatrix}$.

2.  There exist pencils $\bar a(s), \bar b(s)\in{\mathbb F}[s]^{p\times 1}$ and $A_{12}(s)\in {\mathbb F}^{p\times (q-1)}$ such that $ A(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}\bar a(s)&A_{12}(s)\end{bmatrix}$ and $B(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}\bar b(s)&A_{12}(s)\end{bmatrix}$.

-   Let us assume that (i) holds. Then $n\geq \rank(A_{21}(s))\geq \max\{n_1, n_2\}-1\geq n-1$, hence $\rank(A_{21}(s))=n-x$ with $x=0$ or $x=1$. 