### Abstract

We initiate the theory of a quadratic form q over a semiring R, with a view to study tropical linear algebra. As customary, one can writeq(x+y)=q(x)+q(y)+b(x,y), where b is a companion bilinear form. In contrast to the classical theory of quadratic forms over a field, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms q=q_{QL}+ρ, where q_{QL} is quasilinear in the sense that q_{QL}(x+y)=q_{QL}(x)+q_{QL}(y), and ρ is rigid in the sense that it has a unique companion. In case that R is supertropical, we obtain an explicit classification of these decompositions q=q_{QL}+ρ and of all companions b of q, and see how this relates to the tropicalization procedure.

Original language | English |
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Pages (from-to) | 61-93 |

Number of pages | 33 |

Journal | Journal of Pure and Applied Algebra |

Volume | 220 |

Issue number | 1 |

Early online date | 12 Jun 2015 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

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## Cite this

*Journal of Pure and Applied Algebra*,

*220*(1), 61-93. https://doi.org/10.1016/j.jpaa.2015.05.043