This repository demonstrates several examples from my Masters thesis in Mathematics and Statistics at the University of Melbourne. My supervisor for this project was Dr Daniel Murfet.
Part of this thesis involves particular series expansions of polynomials in a completion of the polynomial ring. These series expansions can be computed using polynomial division via an an algorithm referred to as Algorithm 3.3.7 in the thesis. Under some conditions this algorithm will always terminate, which shows that the series expansions are finite and passing to the completion is not necessary. The relevant section of the thesis is Section 3, particularly 3.3 and 3.4. Lemma 3.3.6 precisely states the series expansions we are computing in Algorithm 3.3.7.
In this repository we implement Algorithm 3.3.7 and use it to demonstrate Example 3.4.4 and Example 3.4.5 from the thesis, and an example involving the Dn-type singularity which is mentioned in passing.
Algorithm 3.3.7 is implemented in Python 3 using Sage.
The file examples.py must be run with the version of Python which comes
with Sage. Having installed Sage, run examples.py using the command
sage --python examples.py.