# Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module

## TL;DR

This paper introduces new algorithms, both deterministic and randomized, for computing the characteristic polynomial of a finite rank-two Drinfeld module, improving complexity or applicability over previous methods.

## Contribution

It presents one deterministic and two Monte Carlo algorithms for the problem, expanding computational options and efficiency compared to prior work.

## Key findings

- Algorithms show improved asymptotic complexity
- Algorithms extend parameter space for applicability
- Experimental results validate practical performance

## Abstract

Motivated by finding analogues of elliptic curve point counting techniques, we introduce one deterministic and two new Monte Carlo randomized algorithms to compute the characteristic polynomial of a finite rank-two Drinfeld module. We compare their asymptotic complexity to that of previous algorithms given by Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In particular, we find that all three approaches represent either an improvement in complexity or an expansion of the parameter space over which the algorithm may be applied. Some experimental results are also presented.