Algebraic Geometry Commons^™

The Design And Implementation Of A High-Performance Polynomial System Solver, Alexander Brandt

Electronic Thesis and Dissertation Repository

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.

Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical …

Cache-Friendly, Modular And Parallel Schemes For Computing Subresultant Chains, Mohammadali Asadi

Electronic Thesis and Dissertation Repository

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library.

Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client …

Searching For New Relations Among The Eilenberg-Zilber Maps, Owen T. Abma

Undergraduate Student Research Internships Conference

The goal of this project was to write a computer program that would aid in the search for relations among the Eilenberg-Zilber maps, which relate to simplicial objects in algebraic topology. This presentation outlines the process of writing this program, the challenges faced along the way, and the final results of the project.