A Tropical F5 algorithm

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{\`e}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{\"o}b-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm

Spin-current absorption by inhomogeneous spin-orbit coupling

We investigate the spin-current absorption induced by an inhomogeneous spin-orbit coupling due to impurities in metals. We consider the system with spin currents driven by the electric field or the spin accumulation. The resulting diffusive spin currents, including the gradient of the spin-orbit coupling strength, indicate the spin-current absorption at the interface, which is exemplified with experimentally relevant setups.Comment: 13 pages, 5 figure

Implementation of prime decomposition of polynomial ideals over small finite fields

AbstractAn algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author. To achieve better performance, several improvements are added to the existing algorithm, with strategies for computational flow proposed, based on experimental results. The practicality of the algorithm is examined by testing the implementation experimentally, which also reveals information about the quality of the implementation

Implementation of prime decomposition of polynomial ideals over small finite fields

AbstractAn algorithm for the prime decomposition of polynomial ideals over small finite fields is proposed and implemented on the basis of previous work of the second author. To achieve better performance, several improvements are added to the existing algorithm, with strategies for computational flow proposed, based on experimental results. The practicality of the algorithm is examined by testing the implementation experimentally, which also reveals information about the quality of the implementation

Using the Sum of Roots and Its Application to a Control Design Problem

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