Gröbner free methods and their applications
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Gröbner free methods and their applications
12-14 March 2025
Politecnico di Bari, Bari, Italy
Invited speakers
Per Bäck (Mälardalen University)
Alessandro Oneto (Università di Trento)
The influence of Gröbner bases on Algebra and Geometry during the last decades cannot be overstated; however, using Buchberger's algorithm as a default method in solving problems may lead to unnecessary computations.
Consequently, alternative algorithms have been proposed: Gröbner-free Solving (or, more generally, Degröbnerization) proposes to find alternative ways to get the same solutions, using, for example, tools from Linear Algebra and Combinatorics.
This approach consists in finding new ways to solve specific problems that have been originally solved using Gröbner basis computation and Buchberger's reduction, leaving the use of the latter only to the cases where it is really necessary; usually, the "new ways'' consist in using linear algebra and combinatorial methods.
Degröbnerization is also largely used to study the reverse problem with respect to solving, namely the bonding problem for algebras and ideals:
Given the variety associated with a 0-dimensional ideal, i.e. a finite set of points, the structure of the quotient algebra (which actually contains more information than the ideal itself) can be recovered only using Combinatorics.
Four cornerstones of Degröbnerization are Auzinger-Stetter Matrices, Mourrain's notion of connected to 1, Lundqvist's fast algorithm for merging sorted lists of monomials and adding polynomials and Cerlienco-Mureddu Correspondence.
The aim of this conference is to bring together researchers on the main areas around Degröbnerization. We plan to have a conference, but also a workshop, with a timing sufficiently relaxed to allow participants to discuss and work together.
Topics include
Combinatorial techniques to deal with monomial/polynomial ideals.
Recent developments on the theory -started by Hilbert-on “how to concretely manipulate polynomial ideals”, e.g. ideal theory, resolutions, Hilbert function.
Improvements and/or optimizations of Buchberger's, Janet's and Macaulay's algorithms.
Extensions of these algorithms to a wider class of (not necessarily commutative) rings; for instance, Moeller's reformulation of Buchberger completion/test in terms of his Lifting Theorem is today available in each effectively given ring (in the sense used by Grete Hermann and van der Waerden).
Extensions to subalgebras.
0-dimensional solving/bonding problems.
Gröbner and Gröbner free methods for ideals generated by generic objects.
Application of classical matrices for manipulating algebras and ideals.
Extension of degröbnerization to non-0-dimensional ideals.
Extension of degröbnerization to sub-algebras.
Tag-variable techniques.
Extension of degröbnerization to non-commutative settings.
Parametric polynomial system solving.
Applications, for example to coding theory, cryptography, reverse engineering, biology, algebraic statistics and so on.
The conference will consist of plenary talks by Alessandro Oneto (Università di Trento) and Per Bäck (Mälardalen University), and also contributed talks.
Submission instructions
If you are interested in participating to this event, please send an abstract (3 pages maximum, both in .tex and .pdf) of the talk or problem you want to propose to any of the organizers:
Michela Ceria michela.ceria@gmail.com
Samuel Lundqvist samuel@math.su.se
Teo Mora theofmora@gmail.com
Eduardo Sáenz de Cabezón esaenz-d@unirioja.es
Some of the accepted talks, on decision of the Organizing Committee, will be allowed to submit a complete paper which will be published in a special issue of Journal AAECC, after a referee evaluation according to the standard of the Journal.
Organizing Committee
Michela Ceria (Politecnico di Bari,Italy) michela.ceria@gmail.com
Samuel Lundqvist (Stockholm University, Sweden) samuel@math.su.se
Teo Mora (Università di Genova, Italy) theofmora@gmail.com
Eduardo Sáenz de Cabezón (Universidad de La Rioja, Spain) esaenz-d@unirioja.es
How to reach us
Bari's airport, https://bari.airports.aeroportidipuglia.it/en/
is connected by train to Bari Centrale Station (FNB) https://www.ferrovienordbarese.it/home
which is walking distance to the campus
Programme
Here you can download the programme.
Slides for the talks
Per Bäck From Noncommutative Polynomial Rings to Nonassociative Division Algebras
Fabrice Rouiller Computing parameterizations of zeroes of zero-dimensional Ideals.
Alessandro Oneto 0-dimensional schemes and additive decompositions of polynomials
Francesca Cioffi Cohen-Macaulay, Gorenstein and complete intersection conditions by marked bases
Eduardo Sáenz de Cabezón Involutively stable Ideals and Componentwise Linearity
Problem session: Cioffi
Michela Ceria: On Grobnerian distribution of the escaliers of finite random points.
Anna Torstensson- ErikKennerland: Efficient descriptions of univariate polynomial subalgebras and their initial algebras
Teo Mora: Grobnerian and Grobner Free Techniques on Non-assocative Algebras
Samuel Lundqvist: Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra
Sponsors
Thanks to DMMM (Politecnico di Bari) for the support
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