The past couple of weeks have been interesting. At this point in time, I understand the Sage programming language completely (for the domain of my research), which has allowed for faster progress. I tested the roots that Professor Fastenberg had done earlier, such as the sections when r=1, 2, 3, 7, 10, and 12. All of her roots worked, but since my Sage code differed from Professor Fastenberg’s previous code in Maple, I also discovered other roots. Additionally, by using the solve(…) function in Sage, it allowed me to solve each of the coefficients for the values necessary. The main issue that we encountered at this moment in time is that the roots that Sage puts out in the solve function are approximate. Our goal is to get the exact value, such as in terms of radicals. I tried searching the Sage information database for increased functionality regarding exact roots, but the engine has limits in this regard. Therefore, I need to devise another method to solve for exact roots, probably through a more complex algorithm.
In the upcoming weeks, I will be developing that new algorithm and working on solving for the coefficients of the higher powers of the functions. Another issue that I encountered was that at higher powers, the solve function becomes computationally inefficient. This means that the solver times out, due to the fact that it deals with an insanely large amount of roots, leading to an exponential growth in time.