### Abstract

We show that each algebraic representation of a matroid M in positive characteristic determines a matroid valuation of M, which we have named the Lindström valuation. If this valuation is trivial, then a linear representation of M in characteristic p can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic p if and only if they are linear in characteristic p. To construct the Lindström valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.

Original language | English |
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Pages (from-to) | 688-719 |

Number of pages | 32 |

Journal | Advances in Mathematics |

Volume | 323 |

DOIs | |

Publication status | Published - 7 Jan 2018 |

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### Keywords

- Algebraic matroids
- Matroid valuations

### Cite this

*Advances in Mathematics*,

*323*, 688-719. https://doi.org/10.1016/j.aim.2017.11.006