## Abstract

We develop a theory of Grassmann semialgebra triples using Hasse-Schmidt derivations, which formally generalizes results such as the Cayley-Hamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.

Original language | English |
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Pages (from-to) | 183-201 |

Number of pages | 19 |

Journal | Proceedings of the American Mathematical Society, Series B |

Volume | 7 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020 by the authors.

### Funding

Received by the editors May 14, 2018, and, in revised form, May 13, 2020. 2020 Mathematics Subject Classification. Primary 15A75, 16Y60, 15A18; Secondary 12K10, 14T10. Key words and phrases. Cayley-Hamilton theorem, exterior semialgebras, Grassmann semial-gebras, Hasse-Schmidt derivations, differentials, eigenvalues, eigenvectors Laurent series, Newton’s formulas, power series, semifields, systems, semialgebras, tropical algebra, triples. The first author was partially supported by INDAM-GNSAGA and by PRIN ”Geometria sulle varietà algebriche” Progetto di Eccellenza Dipartimento di Scienze Matematiche, 2018–2022 no. E11G18000350001. The second author was supported in part by the Israel Science Foundation, grant No. 1207/12 and his visit to Torino was supported by the “Finanziamento Diffuso della Ricerca”, grant no. 53 RBA17GATLET, of Politecnico di Torino.

Funders | Funder number |
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INDAM-GNSAGA | E11G18000350001 |

Politecnico di Torino | |

Israel Science Foundation | 53 RBA17GATLET, 1207/12 |

## Keywords

- Cayley-Hamilton theorem
- Grassmann semial-gebras
- Hasse-Schmidt derivations
- Newton’s formulas
- differentials
- eigenvalues
- eigenvectors Laurent series
- exterior semialgebras
- power series
- semialgebras
- semifields
- systems
- triples
- tropical algebra