
Equivalences for Linearizations of Matrix Polynomials
One useful standard method to compute eigenvalues of matrix polynomials ...
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Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods
A polyhedral method to solve a system of polynomial equations exploits i...
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Computing Approximate Common Factors of Matrix Polynomials
Computation of (approximate) polynomials common factors is an important ...
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Holonomic Tools for Basic Hypergeometric Functions
With the exception of qhypergeometric summation, the use of computer al...
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Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors
A fundamental problem in computational algebraic geometry is the computa...
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Wilkinson's bus: Weak condition numbers, with an application to singular polynomial eigenproblems
We propose a new approach to the theory of conditioning for numerical an...
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The Lagrangian remainder of Taylor's series, distinguishes O(f(x)) time complexities to polynomials or not
The purpose of this letter is to investigate the time complexity consequ...
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Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions
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.
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