SCS Undergraduate Thesis Topics
|Javier Vazquez-Trejo||Victor Adamchik||Symbolic Summation in Difference Fields|
We seek to understand a general method for finding a closed form for a given sum that acts as its antidifference in the same way that an integral has an antiderivative. Once an antidifference is found, then given the limits of the sum, it suffices to evaluate the antidifference at the given limits. Several algorithms (by Karr and Schneider) exist to find antidifferences, but the papers describing these algorithms leave out several of the key proofs needed to implement the algorithms. We attempt to fill in these gaps and find that many of the steps to solve difference equations rely on being able to solve two problems: the equivalence problem and the homogenous group membership problem. Solving these two problems is essential to finding the polynomial degree bounds and denominator bounds for solutions of difference equations. We study Karr and Schneider's treatment of these problems and elaborate on the unproven parts of their work.