# Computing representation matrices for the action of Frobenius to cohomology groups

## TL;DR

This paper introduces algorithms for computing Frobenius action matrices on cohomology groups of algebraic varieties over perfect fields, including an efficient method for complete intersections, advancing computational algebraic geometry.

## Contribution

It presents a general algorithm for arbitrary varieties and a specialized, more efficient method for complete intersections, with complexity estimates.

## Key findings

- Algorithm successfully computes Frobenius matrices for various varieties.
- Efficient method significantly reduces computation time for complete intersections.
- Complexity bounds are established for the proposed algorithms.

## Abstract

This paper is concerned with the computation of representation matrices for the action of Frobenius to the cohomology groups of algebraic varieties. Specifically we shall give an algorithm to compute the matrices for arbitrary algebraic varieties with defining equations over perfect fields of positive characteristic, and estimate its complexity. Moreover, we propose a specific efficient method, which works for complete intersections.