Polynomial equations and their solutions are fundamental objects in mathematics and appear across a wide range of applications. As in linear algebra, solutions to equations can be understood geometrically using tools from algebra. The solution sets of nonlinear equations in multiple variables have more interesting geometry and require us to develop new tools. In this class, we will learn how to understand and work with polynomial systems of equations using a mix of algebra, geometry, and algorithms and explore some real-world applications, such as to robotics and computer vision. Topics will include an introduction to polynomial rings and ideals, affine varieties, monomial orderings, Groebner bases, elimination theory, Hilbert's Nullstellensatz, and applications. This class is open to everyone who knows linear algebra and is familiar with mathematical proofs.

Lecture MWF 1:30pm - 2:20pm in IEB 205
Class Syllabus

Instructor: Cynthia Vinzant (email), Office Hours: MW 2:30-3:30pm in PDL C-439
Teaching Assistant: Bryan Lu (email), Office Hours: T 2-3pm, Th 3:30-4:30pm in PDL C-8G

Quick Links: Gradescope, Discussion Board

References
(CLO) Cox, Little, O’Shea. Ideals, Varieties, and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra.
The fourth edition is freely available through the UW libraries, which you are welcome to use. The fifth edition was recently published in 2025, which you are also welcome to use. The portion of the book (Ch. 1-4) we will use for the class are largely the same.

Schedule (Tentative)

Week	Topic	Reading	Homework
March 30 - April 3	Introduction to Polynomials and Varieties	CLO 1.1, 1.2, 1.3	HW 1 (.pdf, .tex), due April 9
April 6 - 10	Ideals, Univariate Polynomials	CLO 1.4, 1.5, 2.1	HW 2 (.pdf, .tex), due April 16
April 13 - 17	Monomial Orderings, Division Algorithm	CLO 2.2, 2.3, 2.4	HW 3, due April 23
April 20 - 24	Groebner Bases	CLO 2.5, 2.6, 2.7	review for midterm
April 27 - May 1	Applications, Midterm 1	CLO 2.8, 3.1	HW 4, due May 7
May 4 - 8	Implicitization and Extensions	CLO 3.2, 3.3, 3.5	HW 5, due May 14
May 11 - 15	The Ideal Variety Correspondence	CLO 4.1, 4.2, 4.3	HW 6, due May 21
May 18 - 22	Irreducible Decomposition	CLO 4.4, 4.5, 4.6	HW 7, due May 28
May 27 - 29 no class May 25	Decompositions and applications	CLO 4.8, 4.9	HW 8, due June 4
June 1 - 5	Applications	CLO 6, Appendix D
June 8	Poster Session 2:30-4:20pm