# Matroids over one-dimensional groups

## TL;DR

This paper develops the theory of matroids over one-dimensional algebraic groups, focusing on positive characteristic, and explores their valuations and duality properties, extending algebraic matroid theory.

## Contribution

It introduces the Lindström valuations and Frobenius flocks for these matroids and shows the class is not closed under duality, advancing algebraic matroid understanding.

## Key findings

- Computed Lindström valuations for these matroids
- Analyzed Frobenius flocks in positive characteristic
- Showed non-closure under duality of algebraic matroids with valuations

## Abstract

We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindstr\"om valuations and Frobenius flocks of such matroids. Building on work by Evans and Hrushovski, we show that the class of algebraic matroids, paired with their Lindstr\"om valuations, is not closed under duality of valuated matroids.