We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$ is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.

中文翻译：

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具有有界秩和固定奇数级的通用对称矩阵多项式

我们确定了奇数级 $d$ 和最多 $r$ 的 $n\times n$ 个复对称矩阵多项式的通用完整特征结构。更准确地说，我们证明了 $n \times n$ 个奇数级 $d$，即度数至多为 $d$，秩至多为 $r$ 的复数对称矩阵多项式的集合是$\lfloor rd/2\rfloor+1$ 对称矩阵多项式集合，具有某些明确描述的完整特征结构。然后，我们证明这些集合在$n\times n$个奇数级$d$且秩至多$r$的复对称矩阵多项式的集合中是开集的。为了证明前面的结果，我们需要推导出具有规定等级、秩和完备本征结构的对称矩阵多项式存在的充要条件，在它们的所有初等因数彼此不同且度数为 $1$ 的情况下。对本文结果的一个重要评论是，在这项工作中识别的通用特征结构与之前工作中识别的具有有界秩和固定等级大于 1 的非结构化和偏对称矩阵多项式的那些完全不同，因为对称的包括特征值，而其他不包括。这种差异需要使用新技术。因为对称的包含特征值而其他的不包含。这种差异需要使用新技术。因为对称的包含特征值而其他的不包含。这种差异需要使用新技术。