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Student Seminar in
Numerical Algebraic Geometry

Description of the course.  Numerical algebraic geometry is a research area that aims to use numerical methods to solve systems of polynomial equations. Sometimes, these numerical algorithms can be turned into theoretical, exact proofs, as we will see. The lectures will cover two major research directions in numerical algebraic geometry: the theory of normal forms and homotopy continuation methods. The latter includes monodromy approaches, zero-dimensional as well as higher-dimensional solution sets, and applications to sampling points on (real) algebraic varieties. Due to time constraints, we will not discuss numerical methods for Riemann surfaces, but we refer the curious reader to [Bobenko, Klein - Computational Approach to Riemann Surfaces].

Structure of the lectures.  There will be 1+6+6 lectures, each of 45+45 minutes, with break in between. The first lecture will be given by C. Meroni as an introduction to the topic. Then, six lectures will cover theoretical aspects in numerical algebraic geometry. Finally, each of the last six lectures will discuss a research paper. There will be 2 presenters per lecture, allowing each student to present both theoretical material and a research paper.

Requirements.  For the six lectures on the theory, the lecturers will need to submit a 2-3 page write-up summarizing the content to be presented. The deadline for submission is 24 hours before the lecture. The write-ups, as well as the slides (if any), will be shared with the other students. Implementations are very welcome and encouraged. For inspiration check the Guides for HomotopyContinuation.jl.

Plan.  These are the topics that we will cover in each lecture:

  1. Introduction and overview: Meroni

  2. Some algebraic geometry to start

  3. Multiplication matrices and more

  4. Homotopy continuation for isolated solutions

  5. Monodromy method

  6. Certification

  7. Positive-dimensional solution sets and sampling 

  8. Guest presentation: Simon Telen  

  9. [TV22] 

  10. [BLOS23] 

  11. [BKSW18] 

  12. [ABH24] 

  13. [BSW24] 

Some references

  • [ABH24] Seth K. Asante, Taylor Brysiewicz, Michelle Hatzel. The algebraic matroid of the Heron variety. arXiv:2401.06286, 2024.

  • [BBCHLS23] Daniel J. Bates, Paul Breiding, Tianran Chen, Jonathan D. Hauenstein, Anton Leykin, and Frank Sottile. Numerical Nonlinear Algebra. arXiv:2302.08585, 2023.

  • [BKSW18] Paul Breiding, Sara Kališnik, Bernd Sturmfels, Madeleine Weinstein. Learning Algebraic Varieties from Samples. Rev. Mat. Complut., 2018.

  • [BLOS23] Paul Breiding, Julia Lindberg, Wern Juin Gabriel Ong, Linus Sommer. Real circles tangent to 3 conics. Le Matematiche, 2023.

  • [BSW24] Paul Breiding, Bernd Sturmfels, Kexin Wang. Computing Arrangements of Hypersurfaces. arXiv:2409.09622, 2024.

  • [BT18] Paul Breiding and Sascha Timme. HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. ICMS 2018. https://www.juliahomotopycontinuation.org

  • [Bry20] Taylor C. Brysiewicz. Newton Polytopes and Numerical Algebraic Geometry. PhD Thesis, Texas A&M University, 2020.

  • [Cox20] David A. Cox. Applications of Polynomial Systems. American Mathematical Society, 2020.

  • [SW05] Andrew J. Sommese and Charles W. Wampler. The Numerical Solution of Systems of Polynomials. World Scientific Publishing Co. Pte. Ltd., 2005.

  • [Tel20] Simon Telen. Solving Systems of Polynomial Equations. PhD Thesis, KU Leuven, 2020.

  • [Tim21] Sascha Timme. Numerical Nonlinear Algebra. TU Berlin, 2021.

  • ​[TV22] ​​Simon Telen, Nick Vannieuwenhoven. A normal form algorithm for tensor rank decomposition. ACM Trans. Math. Software, 2022.

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