Tropical Laurent级数、其tropical根以及非线性矩阵函数特征值的局部化结果
Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions
AI总结：
本文将tropical根理论推广至tropical Laurent级数，刻画其与Newton多边形斜率的对应及无穷多重根性质，并用于经典Laurent级数收敛半径和非线性矩阵特征值局部化。
摘要：
Tropical多项式的tropical根此前已得到研究，并被用于局部化经典多项式的根和矩阵多项式的特征值。我们将tropical根理论从tropical多项式推广到tropical Laurent级数。我们提出的定义确保：与多项式情形一样，tropical根与该tropical Laurent级数相关Newton多边形的斜率之间存在双射。我们表明，与多项式情形不同，tropical根可能有无穷多个；此外，至多有两个tropical根具有无穷多重数。随后，我们通过将经典Laurent级数的内、外收敛半径与其tropicalization的tropical根序列的行为相关联，来应用这一新理论。最后，作为第二个应用，我们讨论了两类局部化结果：一类针对允许局部Laurent级数展开的标量函数的根，另一类针对允许局部Laurent级数展开的正则矩阵值函数的非线性特征值。
英文摘要：
Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.
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。
