Math 648: Computational Algebraic Geometry

Instructor: Frank Sottile
Lectures: TΘ 15:55–17:10 Blocker 121
Course webpage: https://franksottile.github.io/teaching/26.1/648.html
Grading: Some homework/computer projects and end-of-term projects.
Prerequisites: Graduate algebra or permission of instructor. This may be taken concurrently with Math 654.
Geometry seminar Mondays at 15:00 and Fridays at 16:00 in Bloc 302
Texas Algebraic Geometry Symposium, Texas A&M University, 17 – 19 April 2026.
Expected topics to cover:
  • Algebraic-geometric dictionary
  • Resultants and elimination
  • Gröbner bases, including algorithms based on Groebner bases
  • Solving polynomial systems symbolically
  • Some structural properties of algebraic varieties
  • Solving systems of polynomial equations using numerical continuation
  • Certification of numerical solutions. /li>
  • Numerical algebraic geometry. Witness sets and numerical irreducible decomposition
  • Real root counting. Sturm's theorem. Fewnomial theory
  • Toric ideals
  • Toric degenerations and Khovanskii bases
Course description:
    This course will cover the basics of computational algebraic geometry, including the core algorithms in the subject, as well as introduce some of the most common algebraic varieties which occur in applications. We will gain familiarity with software for algebraic geometry, including the systems Macaulay 2, Singular, Bertini, and HomotopyContinuation.jl. Students will complete a final project in the subject which will be presented to the class in lieu of a final exam. Grading will be based on final projects and some written/computer work through the term.
Textook:
There is no textbook, but I will supply some notes that I have developed for the course
The Algebraic-Geometric Dictionary
symbolic computation.
Properties of Varieties.
on Numerical Algebraic Geometry.
You are also welcome to get your hands on the the award-winning Ideals, Varieties, and Algorithms by David Cox, John Little, and Donal O'Shea. In particular it contains all the algebraic preliminaries, and is a great resource. It assumes minimal preprequisites, and may not be completely at the level of an advanced graduate class, but it is well worth reading. It should also be available in a free download from the TAMU library website.
Last modified: Tu Nov 4 9:34:21 CST 2025