summer-drp

This is the Directed Reading Project that ran in Summer 2024, at Brown University. A DRP consists of reading a mathematics book as a group, with a presentation at the end. My group consisted of myself and another undergraduate, and our book of choice was Ideals, Varieties, Algorithms by Cox, Little, and O’Shea. This is a book on computational algebraic geometry at the undergraduate level, and we read through chapters 1-5, 7, and 8. I am working on getting through chapter 9 to round off my understanding of the subject.

My final presentation was on Gröbner bases and Elimination Theory, which is an algorithmic way of solving systems of multivariate polynomial equations. In short, Gröbner bases give us a particularly “nice” way of representing the polynomial consequences of a system of equations- you can think of it as a higher-degree analogue of Gaussian elimination. The first core result of Elimination Theory is the fact that Gröbner basis computation “eliminates” each of the variables except the last, allowing us to determine partial solutions. The second core result characterizes exactly when partial solutions extend to full solutions, giving us a way of extending partial solutions to full solutions.