We investigate rank revealing factorizations of rank deficient $m \times n$ polynomial matrices $P(\lambda)$ into products of three, $P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$, or two, $P(\lambda) = L(\lambda) R(\lambda)$, polynomial matrices. Among all possible factorizations of these types, we focus on those for which $L(\lambda)$ and/or $R(\lambda)$ is a minimal basis, since they allow us to relate easily the degree of $P(\lambda)$ with some degree properties of the factors. We call these factorizations minimal rank factorizations. Motivated by the well-known fact that, generically, rank deficient polynomial matrices over the complex field do not have eigenvalues, we pay particular attention to the properties of the minimal rank factorizations of polynomial matrices without eigenvalues. We carefully analyze the degree properties of generic minimal rank factorizations in the set of complex $m \times n$ polynomial matrices with normal rank at most $r$ and degree at most $d$, and we prove that they are of the form $L(\lambda) R(\lambda)$, where the degrees of the $r$ columns of $L(\lambda)$ differ at most by one, the degrees of the $r$ rows of $R(\lambda)$ differ at most by one, and, for each $i=1, \ldots, r$, the sum of the degrees of the $i$th column of $L(\lambda)$ and of the $i$th row of $R(\lambda)$ is equal to $d$. Finally, we show how these sets of polynomial matrices with generic factorizations are related to the sets of polynomial matrices with generic eigenstructures.

翻译：本文研究秩亏缺的$m \times n$多项式矩阵$P(\lambda)$在三项分解$P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$或两项分解$P(\lambda) = L(\lambda) R(\lambda)$下的秩揭示分解问题。在所有此类分解中，我们重点关注$L(\lambda)$和/或$R(\lambda)$为极小基的情形，因为这种分解能方便地将$P(\lambda)$的次数与因子的次数性质关联起来。我们将此类分解称为极小秩分解。受"复域上秩亏缺多项式矩阵在一般情形下无特征值"这一经典事实的启发，我们特别关注无特征值多项式矩阵的极小秩分解性质。我们系统分析了正常秩不超过$r$、次数不超过$d$的复$m \times n$多项式矩阵集合中一般极小秩分解的次数特性，并证明其具有$L(\lambda) R(\lambda)$形式，其中$L(\lambda)$的$r$个列的次数至多相差1，$R(\lambda)$的$r$个行的次数至多相差1，且对每个$i=1, \ldots, r$，$L(\lambda)$第$i$列与$R(\lambda)$第$i$行的次数之和等于$d$。最后，我们揭示了具有一般分解的这类多项式矩阵集合与具有一般特征结构的多项式矩阵集合之间的关联。