Matías R. Bender

Since 2023, I have been working as a researcher (chargé de recherche) at the project/team Tropical at INRIA Saclay – Île-de-France - CMAP, École polytechnique in Palaiseau, France. Prior to this, I was a Postdoctoral researcher in the Algorithmic Algebra group at the Institut für Mathematik, Technische Universität Berlin, led by Peter Bürgisser. In June 2019, I earned my PhD from Sorbonne Université, under the guidance of Jean-Charles Faugère and Elias Tsigaridas.

My research focuses on polynomial systems, where I study algorithms for solving them and their applications in various domains. My PhD thesis specifically dealt with effective algorithms for solving sparse polynomial systems using Gröbner bases and Resultants. Take a look!

Publications

• Solving bihomogeneous polynomial systems with a zero-dimensional projection
  Matías R. Bender, Laurent Busé, Carles Checa, and Elias Tsigaridas.
  Proceedings of the 50th International Symposium on Symbolic and Algebraic Computation, 2025. To appear.
  ArXiv version.
• Dimension results for extremal-generic polynomial systems over complete toric varieties
  Matías R. Bender and Pierre-Jean Spaenlehauer
  Journal of Algebra, 2024.
  ArXiv version.
• Solving sparse polynomial systems using Groebner bases and resultants
  Proceedings of the 47th International Symposium on Symbolic and Algebraic Computation, 2022. In press.
  Invited paper complementing tutorial.
  ArXiv version.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Matías R. Bender and Simon Telen
  Mathematics of Computation - AMS, 2022.
  ArXiv version.
• Koszul-type determinantal formulas for families of mixed multilinear systems
  Matías R. Bender, Jean-Charles Faugère, Angelos Mantzaflaris, and Elias Tsigaridas
  SIAM Journal on Applied Algebra and Geometry, 2021.
  ArXiv version, MATLAB code
• Classifier Construction in Boolean Networks Using Algebraic Methods
  Robert Schwieger, Matías R. Bender, Heike Siebert, and Christian Haase
  Computational Methods in Systems Biology, Lecture Notes in Computer Science, 2020.
  ArXiv version, Python code
• A nearly optimal algorithm to decompose binary forms
  Matías R. Bender, Jean-Charles Faugère, Ludovic Perret, and Elias Tsigaridas
  Journal of Symbolic Computation, 2021.
  HAL version
• Groebner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems
  Matías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas
  Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation, 2019
  HAL version
• Towards Mixed Groebner Basis algorithms: the Multihomogeneous and Sparse case
  Matías R. Bender, Jean-Charles Faugère, and Elias Tsigaridas
  Proceedings of the 43th International Symposium on Symbolic and Algebraic Computation, 2018
  HAL version
• Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors
  Matías R. Bender, Jean-Charles Faugère, Angelos Mantzaflaris, and Elias Tsigaridas
  Proceedings of the 43th International Symposium on Symbolic and Algebraic Computation, 2018
  HAL version, Examples and data, Video.
• A Superfast Algorithm to Decompose Binary Forms
  Matías R. Bender, Jean-Charles Faugère, Ludovic Perret, and Elias Tsigaridas
  Proceedings of the 41th International Symposium on Symbolic and Algebraic Computation, 2016
  SIGSAM's ISSAC Distinguished Student Author Award 2016
  HAL version

Preprints

• Multigraded Castelnuovo-Mumford regularity and Gröbner bases
  Matías R. Bender, Laurent Busé, Carles Checa, and Elias Tsigaridas
  Submitted, 2024. (25 pages)
• Efficient computation of multiparameter persistence
  Matías R. Bender, Oliver Gäfvert, and Michael Lesnick
  In preparation. (48 pages)
• Yet another eigenvalue algorithm for solving polynomial systems
  Matías R. Bender and Simon Telen
  Accepted, May 2021. (28 pages)
  Julia package EigenvalueSolver.jl

Software

• Muphasa (Joint work with Oliver Gäfvert, and Michael Lesnick)
  Open-source C++ software (~6k lines) to compute presentations of multi-parameter persistent homologies.
• EigenvalueSolver.jl (Joint work with Simon Telen)
  Open-source Julia package (~1k lines) to compute solve numerically polynomial systems.
  More information.
• sylvesterMEP (Joint work with Jean-Charles Faugère, Angelos Mantzaflaris, and Elias Tsigaridas)
  Open-source Matlab script to solve the multi-parameter eigenvalue problem.
  More information.
• Classifier-construction (Joint work with Robert Schwieger, Heike Siebert, and Christian Haase)
  Open-source python code to compute classifiers on Boolean networks using Groebner basis.
  More information.

Students

PhD students

• Chaoping Zhu (coadvisor Elias Tsigaridas). Sorbonne Université, Paris, France. Since 2024.

Master and bachelors students

• Rita Arfoul (M2 thesis, coadvisor Elias Tsigaridas). Aix-Marseille University, France. 2024.
  After: PhD student at Université du Littoral Côte d'Opale, France.
• Tom Baumbach (M2 thesis). TU Berlin, Berlin, Germany. 2022.
  After: PhD student at TU Berlin (advisor Martin Henk).
• Tom Baumbach (Bachelor thesis). TU Berlin, Berlin, Germany. 2022.
  His thesis was awarded the BMG-Bachelorpreise by Berliner Mathematische Gesellschaft e.V.

Talks

• Solving Polynomial Systems Using Determinantal Formulas
  International Workshop on Recent Advances in Mathematics (RAMA 2024)
  Beihang University, Hangzhou, China, November 2024.
• Computing minimal presentations of multi-parameter persistent homology (contributed)
  4th workshop on Computational Persistence
  TU Graz, Graz, Austria, September 2024.
• Solving Polynomial Systems Using Determinantal Formulas
  Talk at session at SIAM Conference on Applied Linear Algebra (LA24)
  Paris, France, May 2024.
• A new symbolic-numeric method to solve the multiparameter eigenvalue problem
  Back to the roots' seminar talk
  Department of Electrical Engineering, KU Leuven, Leuven, Belgium, March 2024.
• Multigraded Castelnuovo-Mumford regularity and Groebner bases
  MAX team's seminar
  Computer Sciences department, École Polytechnique, Palaiseau, France, March 2024.
• Introducción amigable al álgebra no lineal y sus aplicaciones (in Spanish)
  Talk at session at 10th International Congress on Industrial and Applied Mathematics (ICIAM2023)
  (Online) Universidad Autónoma de Chiapas, Chiapas, Mexico, November 2023.
• Dimension Results for Polynomial Systems Over Complete Toric Varieties
  Talk at session at 10th International Congress on Industrial and Applied Mathematics (ICIAM2023)
  Tokyo, Japon, August 2023.
• Dimension Results for Polynomial Systems Over Complete Toric Varieties
  Talk at session at SIAM Conference on Applied Algebraic Geometry (AG23)
  Eindhoven, The Netherlands, July 2023.
• Introductory course Computational Commutative Algebra
  CIMPA school Algebraic and Tropical Methods for Solving Differential Equations
  UNAM, Oaxaca, Mexico, June 2023.
• Groebner bases and systems of sparse polynomial equations
  Seminar Algebra and Discrete Mathematics
  (Online) Johannes Kepler University, Linz, Austria, April 2023.
• Symbolic-numeric methods to solve polynomial systems
  Workshop on Solving Polynomial Equations and Applications
  CWI, Amsterdam, The Netherlands, October 2022.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Tropical team's seminar
  CMAP, École polytechnique, Palaiseau, France, April 2022.
• Polynomial systems, sparsity, and applications
  Caramba team's seminar
  (Online) Inria Nancy - Grand Est, Nancy, France, February 2022.
• Polynomial systems, sparsity, and applications
  Ouragan team's seminar
  (Online) INRIA Paris, Paris, France, February 2022.
• Polynomial systems, sparsity, and applications
  AROMATH team's seminar
  (Online) INRIA - Université Côte d'Azur, Sophia Antipolis, France, February 2022.
• Polynomial systems, sparsity, and applications
  Brainstorming days on measure and polynomial optimization
  (Online) Laboratoire d'analyse et d'architecture des systèmes, Toulouse, France, November 2021.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Mini-symposium New Trends in Polynomial System Solving
  SIAM Conference on Applied Algebraic Geometry
  Online, August 2021.
• Koszul-type determinantal formulas for families of mixed multilinear systems
  Seminar on Non-Linear Algebra
  MPI für Mathematik in den Naturwissenschaften, Leipzig, Germany, August 2021.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Applied Algebra and Analysis Online Seminar
  (Online) TU Braunschweig and University of Osnabrueck, Germany, April 2021.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Journées Nationales de Calcul Formel 2021
  (Online) CIRM, Luminy, France, March 2021
  Video, Slides.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  AROMATH team's seminar
  (Online) INRIA, Sophia Antipolis, France, December 2020
• Gröbner bases and sparse polynomial systems
  Session: Gröbner Bases in Theory and Practice
  (Online) International Congress on Mathematical Software 2020, July 2020.
• Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
  Kolloquium on Algorithmic Algebra at TU Berlin
  (Online) Technische Universität Berlin, Berlin, Germany, July 2020.
• Solving sparse polynomial systems by computing Gröbner bases faster
  Seminar on Non-Linear Algebra
  MPI für Mathematik in den Naturwissenschaften, Leipzig, Germany, March 2020
• Solving sparse polynomial systems by computing Gröbner bases faster
  Seminar of the Institut für Algebra
  Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany, January 2020
• Algorithms for sparse polynomial systems: Gröbner bases and resultants
  Kolloquium on Algorithmic Algebra at TU Berlin
  Technische Universität Berlin, Berlin, Germany, October 2019
• Fórmulas determinantales para el resultante de algunos sistemas multilineales mixtos
  Departamento de Matemática, FCEN, UBA.
  Buenos Aires, Argentina, September 2019
• Groebner Basis over Semigroup Algebras: Algorithms and Applications for Sparse Polynomial Systems
  44th International Symposium on Symbolic and Algebraic Computation
  Beijing, China, July 2019
• Algorithms for sparse Groebner basis
  Mini-symposium Symbolic-numeric methods for non-linear equations: Algorithms and applications at SIAM Conf. on Applied Algebraic Geometry
  Bern, Switzerland, July 2019
• Solving sparse polynomial systems using Groebner bases
  Summer School on Geometric and Algebraic Combinatorics
  Sorbonne Université, Paris, France, June 2019
• Determinantal formulas for the resultant of some mixed multilinear systems
  Structured Matrix Days 2019
  XLIM-DMI, Université de Limoges, Limoges, France, May 2019.
  Slides
• Solving sparse polynomial systems using Groebner bases
  GRAPE kick-off meeting
  IMBM Bogazici University, Istanbul, Turkey, May 2019
• Solving sparse polynomial systems using Groebner bases
  CARAMBA team's seminar
  INRIA, Nancy, France, April 2019
• Solving Polynomial Systems
  Inria's Junior Seminar
  INRIA, Paris, France, February 2019
• Solving sparse polynomial systems using Groebner bases
  Facets of Complexity Seminar
  Freie Universität Berlin, Berlin, Germany, February 2019
• Towards Mixed Groebner Basis algorithms: the Multihomogeneous and Sparse case
  43th International Symposium on Symbolic and Algebraic Computation
  The City University of New York, New York, NY, United States of America, 17-19 July 2018
• Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors
  43th International Symposium on Symbolic and Algebraic Computation
  The City University of New York, New York, NY, United States of America, 17-19 July 2018
• A Superfast Algorithm to Decompose Binary Forms
  41th International Symposium on Symbolic and Algebraic Computation
  Wilfrid Laurier University, Waterloo, Ontario, Canada, 20-22 July 2016
• Decomposition of Binary Forms
  GOAL Workshop in Paris 2016
  LIP6, UPMC, Paris, France 20-21 May 2016
• A Superfast Algorithm to Decompose Binary Forms
  Structured Matrix Days 2016
  XLIM-DMI, Université de Limoges, Limoges, France, 9-10 May 2016

Education

• PhD Student, Sorbonne Université, Paris, France. 2016 - 2019.
  Advisors: Jean-Charles Faugère and Elias Tsigaridas (PolSys - LIP6 - SU/INRIA/CNRS)
  Title: Algorithms for sparse polynomial systems: Gröbner basis and resultants
  Manuscript
• Licenciatura en Ciencias de la Computación, Universidad de Buenos Aires, Buenos Aires, Argentina. 2010 - 2015
  (Equivalent to Master's degree in Computer Science, 6-years studies with a final thesis)

Awards, Grants, and Recognitions

• PGMO grant (12.5€). PI at the project SOAP (Sparsity in Optimization via Algebra and Polynomials) funded by Fondation Mathématique Jacques Hadamard. 2023-2025.
• Invited tutorial at the 47th International Symposium on Symbolic and Algebraic Computation . Lille, France. 2022.
• SIAM Early Career Travel Award to attend SIAM Conference on Applied Algebraic Geometry . 2019.
• Distinguished Student Author Award. ACM's SIGSAM. 2016.
  For the paper "A Superfast Algorithm to Decompose Binary Forms", published at ISSAC 2016
• PhD Fellowship. INRIA. 2015.
  Three years funding (CORDI-S)
• Academic Merit Award. Santander Río Bank. 2015.
• Research Scholarship. INRIA - Campus France. 2015.
  Research intership at PolSys.
• Exchange Scholarship to Universidad Autónoma de Madrid (UAM), Spain. Santander Río Bank. 2014.
  Fully paid semestral exchange.
• Research Fellowship “Beca Estímulo a las Vocaciones Científicas 2013”. Consejo Interuniversitario Nacional (CIN), Argentina. 2013.
  Under the supervision of Esteban Feuerstein.

Teaching and Popular Science

Teaching

• Chargé d'enseignement, Computer Science Department, École Polytechnique. Since September 2023
  TDs in Advanced Programming and Computer Programming from the bachelor program in CS.
• Introductory course Computational Commutative Algebra (5hs)
  CIMPA school Algebraic and Tropical Methods for Solving Differential Equations
  UNAM, Oaxaca, Mexico, June 2023.
• Teaching Assistant, Computer Science Department, University of Buenos Aires. March 2012 - February 2014
  TA at Algorithms & Data Structures II, two semesters in 2012 and one in 2013. Topics include formal specification using ADTs, Software Design, Data Structures, Sorting Algorithms, Divide and Conquer techniques, Complexity Order calculations. Implementations with C++.
  TA at Algorithms & Data Structures III, one semester in 2013. Topics include Graph Theory, Complexity Theory (P and NP), Dynamic Programming and Backtracking techniques.
• Informatics Teacher, Escuela Integral Hebrea Natan Gesang, (primary school). April 2010 - January 2012
  Computer teacher for kids from six to eight years. Topics include different OSs, basic ofimatics, web design (DreamWeaver) and elementary programming (Scratch).

Popular Science    (Click to see)

• Online Tutor at Dale Aceptar, Foundation “Dr. Manuel Sadosky”. Dale Aceptar is an educational project by the Foundation “Dr. Manuel Sadosky” to encourage high school students to pursue IT careers. The participants learn how to program in Alice, an educational programming language to make games and animations, and how to program a chatterbot using Chatbot, developed by the FaMAF, UNC. My role was to provide guidance to students and tutor them through the project. October 2012 - August 2013
• Collaborator at the workshop “Jornadas Program.ar”. Two-day long workshop aimed to high-school students about programming in Alice. Organized by the Chief of the Cabinet of Ministers, Foundation “Dr. Manuel Sadosky” and Ministry of Education. August 2013.
• Lecturer at the programming workshop of the Computer Science Week 2013 organized by the Computer Science Department, FCEN-UBA. August 2013.
• Speaker at the two inauguration talks for the new students of the Computer Science department at the beginning of each semester of 2013. Organized by the Vocational Orientation Secretary (DOV), FCEN-UBA.
• Designer, planning and lecturer at an eight-class long workshop of the programming language Alice destined to high-school students. Organized by the Vocational Orientation Secretary (DOV), FCEN-UBA. August 2012 - September 2012
• Expositor at Tecnópolis. Promotion of the Foundation “Dr. Manuel Sadosky” project “Dale Aceptar ”, at Tecnópolis, the national exhibition of science, technology, industry and arts. My tasks included recruiting high school students to join the project and giving them tutoring about the Alice programming language. July 2012 - September 2012
• Expositor at Computer Science Week 2012. Graph theory stand. September 2012.
• Collaborator at Mathematics Week 2012, organized by the Mathematics Department, FCEN-UBA. April 2012.
• Designer and lecturer at a three-class long Alice’s workshop aimed to CBC students, as part of the program PACENI-Exactas, FCEN-UBA. June 2012.
• Collaborator at the two first workshops “Introduction to programing with 3D worlds” organized by the Foundation “Dr. Manuel Sadosky” and Conectar Igualdad. November 2012 - December 2012.
• Teacher at the project En la Tecla. This project is a computer literacy program destined to people with low resources, mainly from the Slum 31 (Villa 31). August 2011 - December 2011.
• Expositor at Computer Science Week 2011. Logics stand. September 2011.
• Collaborator at Mathematics Week 2011. April 2011.

Administrative responsabilities

• Inria Saclay's scientific commission member, Palaiseau, France. Since 2023.
  This commission selects recipients of PhD and postdoc grants funded by the institute.
• Publicity chair at International Symposium on Symbolic and Algebraic Computation 2025 (ISSAC'25). Guanajuato, Mexico, 2025.
• Award comittee member for best software at ISSAC'19. Beijing, China, 2019.
• Students' councilor, Council of the Computer Science Department (CoDep), DC-FCEN-UBA.
  February 2013 - January 2014. Elected by the students.

Reach me