# Algebraic matroids and Frobenius flocks

## TL;DR

This paper introduces the Lindström valuation linking algebraic and linear matroid representations in positive characteristic, and shows that rigid matroids are algebraic if and only if they are linear in that characteristic.

## Contribution

It defines the Lindström valuation for algebraic matroid representations and introduces flock representations to connect algebraic and linear matroids in positive characteristic.

## Key findings

- Rigid matroids are algebraic iff they are linear in the same characteristic.
- Algebraic representations induce matroid valuations called Lindström valuations.
- Flock representations are a new tool for representing algebraic matroids.

## 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 {\em Lindstr\"om 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 Lindstr\"om valuation, we introduce new matroid representations called flocks, and show that each algebraic representation of a matroid induces flock representations.