About me

I am a postdoc in the Pascaline team. My topics of interest lie in computer algebra, numerical algebraic geometry and topology. Precisely, I focus on certified homotopy continuation and applications.

Previously, I was a PhD student in the Mathexp team, under the supervision of Pierre Lairez. I defended my PhD on September 9th, 2026. [Manuscript][Slides]

Preprints and publications

• Computing Braids from Approximate Data. [arXiv]
  Alexandre Guillemot and Pierre Lairez.
  Preprint. January 2026.
  
  Abstract

  We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under numerical uncertainty. Instead, we formalize an input model for approximate data, based on a separation predicate. It applies, for example, to paths obtained by tracking the roots of a parametrized polynomial with complex coefficients, thereby connecting certified path tracking outputs to exact braid computation.
  
  
• Belyi map verification using certified path tracking. [arXiv]
  Alexandre Guillemot and John Voight.
  To appear. April 2026.
  Abstract

  We provide an end-to-end workflow to rigorously compute the monodromy of Belyi maps from exact equations over number fields using certified homotopy continuation. We then apply this method at scale to certify the monodromy triples of Belyi maps in the $L$-functions and Modular Forms Database (LMFDB).
  
  
• Certified Algebraic Path Tracking with Algpath. [doi] [hal] Issac '25 distinguished software presentation!
  Alexandre Guillemot.
  ACM Commun. Comput. Algebra 59.3 (Jan. 2026), pp. 53–56.
  
  Abstract

  Algpath is a certified homotopy continuation software. We upgrade the previous fixed-precision Rust implementation by incorporating mixed, adaptive precision with minimal overhead. This allows us to tackle problems on which the initial implementation fails due to the inability to increase precision, and where uncertified methods may fail or path jump.
  
  
• Validated numerics for algebraic path tracking. [doi] [hal] [arXiv]
  Alexandre Guillemot and Pierre Lairez.
  Proc. ISSAC 2024. New York, NY, USA: Association for Computing Machinery, July 2024, pp. 36–45.
  
  Abstract

  Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.
  
  

Software

• algpath, joint implementation with Pierre Lairez
  
  About

  algpath is a Rust package for the rigorous computation of the continuation of a regular zero of a parametrized polynomial system as the parameter varies.
  
  
• algpath bench
  
  About

  algpath bench is a repository for homotopy continuation benchmarking. It is able to test different packages on polynomial systems, and to process the results in the form of tables and latex output.
  
  
• braids
  
  About

  braids is a Rust package for the computation of braids induced by the displacement of the roots of a parametrized polynomial with complex coefficients. It relies on algpath for the certified path tracking step.
  
  
• Belyi monodromy
  
  About

  Belyi Monodromy is a repository for the rigorous computation of the monodromy triples of Belyi maps listed in the LMFDB, using the certified path tracking software Algpath. As a byproduct, it provides utilities for computing the monodromy of Belyi maps in general.
  
  

Talks

• ICMS 2026, Wilfrid Laurier University and University of Waterloo, July 2026. [Slides]
• Journées Nationales de Calcul Formel (JNCF), CIRM, March 2026. [Slides]
• Ouragan seminar, imj-prg, December 2025. [Slides]
• Journées de géométrie algorithmique (Jga), station biologique de Roscoff, October 2025. [Slides]
• Pascaline seminar, LIP, ENS Lyon, September 2025. [Slides]
• International Symposium on Symbolic and Algebraic Computation (ISSAC), Center for Research in Mathematics (CIMAT), July-August 2025. [Slides]
• Applications of Computer Algebra (ACA), Cultural Conference Center of Heraklion, July 2025. [Slides]
• SIAM Conference on Applied Algebraic Geometry (AG), University of Wisconsin Madison, July 2025. [Slides]
• Séminaire AG, LMV, May 2025. [Slides]
• Journées Nationales de Calcul Formel (JNCF), CIRM, March 2025. [Slides]
• International Symposium on Symbolic and Algebraic Computation (ISSAC), NC State University, July 2024. [Slides]
• Numerical Safety for Computer-Aided Proofs (NUSCAP), LIP6, May 2024. [Slides]
• Journées Nationales de Calcul Formel (JNCF), CIRM, March 2024. [Slides]

Teaching

• CSC_41011_EP - Les bases de la programmation et de l'algorithmique (teaching assistant, two groups), École polytechnique, fall 2025.
• CSC_1S002_EP - Computer Programming (teaching assistant, one group), École polytechnique, summer 2024.
• CSC_41011_EP - Les bases de la programmation et de l'algorithmique (teaching assistant, two groups), École polytechnique, fall 2024.
• MAA101 - Linear Algebra (teaching assistant, two groups), École polytechnique, fall 2023.