Algorithms for finding the gradings of reduced rings

TL;DR

This thesis introduces an algorithm to compute the universal grading of reduced rings, extending previous results and providing a practical method with specific runtime complexity.

Contribution

It presents the first algorithm for computing the universal grading of reduced rings and generalizes the concept to a broader class of rings.

Findings

Algorithm computes universal grading with runtime n^{O(m)}.

Successfully generalizes universal grading to more rings.

Provides a method to compute all gradings of the associated algebra.

Abstract

This work is a Master thesis supervised by Prof. Dr. H.W. Lenstra. Lenstra and Silverberg showed that each reduced order has a universal grading, which can be viewed as the `largest possible grading'. We present an algorithm to compute the universal grading for a given order $R$, which has runtime $n^{O(m)}$, where n is the length of the input and m is the size of the minimal spectrum of $R$. We do this by computing all gradings of the corresponding reduced $\mathbb{Q}$-algebra with cyclic abelian groups of prime-power order. We additionally generalize the result of Lenstra and Silverberg that reduced orders have a universal grading to a broader class of rings.