We consider $$m \times s$$ m × s matrices (with $$m\ge s$$ m ≥ s ) in a real affine subspace of dimension n . The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in $$\left( {\begin{array}{c}n+m(s-r)\\ n\end{array}}\right) $$ n + m ( s - r ) n . It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach.

中文翻译：

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低秩线性矩阵的实根求法

我们在维度为 n 的实仿射子空间中考虑 $$m \times s$$ m × s 矩阵（具有 $$m\ge s$$ m ≥ s ）。在这样的空间中寻找低秩元素的问题在信息和系统理论中有许多应用，其中低秩是结构和简约的同义词。我们设计了计算机代数算法，基于多项式系统求解的先进方法，以高效准确地解决这个问题：输入是跨越仿射子空间的矩阵的有理系数以及期望的最大秩，输出是有理数参数化编码与低秩实代数集的每个连通分量相交的有限点集。我们对算法的复杂性进行了彻底的研究。它是 $$\left( {\begin{array}{c}n+m(sr)\\ n\end{array}}\right) $$ n + m ( s - r ) n 的多项式。它改进了最先进的计算机代数和有效的实代数几何。此外，计算机实验显示了我们方法的实际效率。