矩阵的可积性
Integrability of matrices
AI总结：
本文解决了矩阵可积性的一般问题，通过特征多项式给出了矩阵可积的充要条件，并基于Shabat多项式等特殊多项式性质确定了特定特征值几何重数下可积与不可积矩阵的存在条件。
摘要：
矩阵的微分与积分概念已知。尽管每个矩阵都可微，给定矩阵是否可积尚不明确。近期对角化矩阵取得了一些进展，但一般问题仍未解决。本文给出了可积性问题的完整解答，即用特征多项式给出了矩阵可积的充要条件。此外，我们还找到了具有给定特征值几何重数的可积与不可积矩阵存在的充要条件。我们的方法依赖于数论与动力学中Shabat多项式和保守多项式等特殊多项式类的性质。
英文摘要：
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
