Welcome to the Weekly Seminars on Algorithms in Algebraic Geometry taking place on Mondays 14:00-15:30 CET, at KU Leuven campus Arenberg 3, in the  seminar room B02.18 (full name 200B.02.018; second floor of the Math building 200B).

Organizers: Rémi Prébet and Erdenebayar Bayarmagnai

Exceptionally, the session of April 29th is moved to 15:30-17:00 CET in room 200B.02.014

Registration:

Please email Rémi if you are interested in attending the meeting or have any questions.

Overview:
The main goal of this reading seminar is to get familiar with some algorithmic aspects of both classic and real algebraic geometry. The program will be divided in two parts. We will first survey various techniques on complex algebraic varieties, before switching to their real counterpart. These will be introduced and illustrated with several examples, using both hands and computer algebra softwares.

Current 

Schedule:

• February 19, 2024 - B02.18: Rémi Prébet
  Introduction and overview: After recalling some basic notions both from complex and real algebraic geometry, I will present and use some classic algorithmic tools to perform basic operations on algebraic varieties and solve geometric problems both from pure and applied side.
  
  

• February 26, 2024 - B02.18: Erdenebayar Bayarmagnai
  Introduction to Gröbner bases: In this talk, I will talk about monomial orderings, a division algorithm and Gröbner bases, and explain how Gröbner bases solve the ideal membership problem.
  
  

• March 4, 2024 - B02.18: Ngoc-Anh Nguyen
  Applications of Gröbner bases: I will discuss some of the applications of Gröbner basis, particularly for consistency problems, radical membership testings, and some geometric operations of algebraic sets (with the help of elimination theorem).
  
  

• March 11, 2024 - B02.18: Sebastian Seemann
  Dimension of Varieties: In this talk we will discuss dimension of Varieties and give a fully algebraic and implementable definition. We start by studying monomial Ideals and their complements and finish with a study of affine Hilbert functions. Then we will see how Gröbner basis theory allows us to apply Hilberts results for monomial Ideals to arbitrary Ideals. 
  
  

• March 18, 2024 - S.01.05: Giacomo Masiero
  The F4 algorithm: The F4 algorithms are a family of algorithms which improve in efficiency the classical Buchberger’s method to compute Gröbner bases for polynomial ideals. In the first part, we will investigate their underlying philosophy, which is inspired by homogeneous Gröbner bases and linear algebra. In the second part, we will highlight how these reflect in the pseudocode of the F4 algorithms.
  
  

• March 25, 2024 - B.00.05:  Cas Proost
  The Groebner fan and computational tropical geometry:  In this talk, I will define tropical varieties from the viewpoint of groebner theory and polyhedral  geometry, starting from the Groebner fan.
  
  

• April 15, 2024 - B01.18:  Emiliano Liwski
  Quantifier Elimination Over Algebraically Closed Fields: In this talk I will give the definition of constructible sets and I will prove that, in the context of algebraically closed fields, the projection of a constructible set is constructible.
  
  

• April 22, 2024 - B02.18: Erdenebayar Bayarmagnai
  Real root counting:  I will talk about two main results of counting real roots of univariate polynomials with real coefficients. The first result is Budan-Fourier theorem which gives an  upper bound for the number of real roots counted with multiplicities in (a,b). The second result is Sturm's  theorem which exactly counts the number of real roots without multiplicities in (a,b).
  
  

• April 29, 2024 - B02.14 (15:30-17:00) : Rémi Prébet
  Quantifier Elimination over real numbers: After some reminders of the previous sessions, we will define quantifier elimination when dealing with real numbers by analogy with the algebraically closed case we saw previously. Following the analogy, this will lead us to define a new class of sets, namely semi-algebraic sets, and which is stable by projection. The proof of the latter will be at the core of the presentation, and will heavily rely on the techniques introduced in the two previous talks.
  
  

• May 6, 2024 - B02.18: Ngoc-Anh Nguyen
  Cylindrical Algebraic Decomposition (CAD) and applications: TBA
  
  

• May 13, 2024 - B02.18: Sebastian Seemann
  Subresultant sequences : TBA
  
  

• May 20, 2024 - B02.18: Giacomo Masiero
  Computing Remainder and Subresultant sequences: TBA
  
  

• May 27, 2024 - B02.18: Emiliano Liwski
  Cylindrical Algebraic Decomposition Algorithm and applications: TBA
  
  

Books:

• Ideals, Varieties and Algorithms by David A.Cox, John Little and Donal O'Shea

• Algorithms in Real Algebraic Geometry by Saugata Basu, Richard Pollack, and Marie-Françoise Roy