$H(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}
  h(s)\\H_1(s)\end{bmatrix}$ if and only if $$\label{theta}
\bar \theta \geq \theta,$$ $$\label{inter}
\pi_i(s, t)\mid\pi^1_i(s, t)\mid\pi_{i+1}(s, t), \quad 1\leq i \leq n,$$ $$\label{row}
{\mathbf u}\prec' {\mathbf w},$$ $$\label{colequal}
{\mathbf g}={\mathbf c}.$$

Next, we state the result when $x=1$, $y=0$.

\[\] \[lemmaDox1\] Let $H_1(s)\in {\mathbb F}[s]^{(n+p) \times (n+m)}$, $H(s)\in {\mathbb F}[s]^{ (n+p+1)\times (n+m)}$ be matrix pencils of $\rank(H_1(s))=n$, $\rank(H(s))=n+1$.

Let $\pi^1_1(s, t)\mid \dots \mid \pi^1_{n}(s, t)$, $g_1 \geq \dots \geq g_{m}\geq 0$, and $w_1 \geq \dots   \geq w_{p}$ be the homogeneous invariant factors, the column and the row minimal indices of $H_1(s)$, respectively, and let $\pi_1(s, t)\mid \dots \mid \pi_{n+1}(s, t)$, $c_1 \geq \dots \geq c_{m-1}\geq 0$, and $u_1 \geq \dots  \geq u_{p}$ be the homogeneous invariant factors, the column and the row minimal indices of $H(s)$, respectively.

Let ${\mathbf g}=(g_1, \dots, g_m)$, ${\mathbf w}=(w_1, \dots, w_p)$, ${\mathbf c}=(c_1, \dots, c_{m-1})$, and ${\mathbf u}=(u_1, \dots, u_{p})$.

There exists a pencil $h(s)\in{\mathbb F}[s]^{1 \times (n+m)}$ such that $H(s){\ensuremath{\stackrel{s.e.}{\sim}}}\begin{bmatrix}
  h(s)\\H_1(s)\end{bmatrix}$ if and only if (\[inter\]), $$\label{col1}
{\mathbf g}\prec' {\mathbf c},$$ $$\label{rowequal1}
{\mathbf w}={\mathbf u}.$$

Technical results {#sectechnical}
=================

The next results are technical lemmas about 1step-generalized majorizations. All of the sequences they deal with are ordered partitions of nonnegative integers.

\[propbat\] Let $S\geq 0$ 