ACA 2021

SPECIAL SESSION at the 26TH International Conference on Applications of Computer Algebra (ACA2021)

July 23-27, 2021, Virtual, Online

Hilbert's and Gordan's proofs of the Basissatz, which both consisted in giving an algorithm (an algorithm, not a procedure!) for producing a Groebner basis of a given ideal, settled the elementary approach toward Buchberger Theory: from the need of a term-ordering (stated implicitly by Hilbert and explicitly by Gordan) to the introduction of a rewriting procedure, which in Gordan is exactly Buchberger's.

The first group of researchers who deeply studied the notions and the tools introduced by Hilbert in his seminal paper, as Macaulay, Gunther and (mainly) Janet (who, studying with and under Hilbert, reinterpreted Riquier's results) are, consequently, the first which introduced the most important alternatives to Buchberger Algorithm for producing Groebner bases: Macaulay's Matrix from which Faugère's F4-F5 stemmed and Janet's involutive bases.

Recent research oriented itself toward the following three lines of investigation, which also represent the main focus of the talks expected for this section:

• to improve and optimize Buchberger's, Janet's and Macaulay's algorithms;

• to extend them 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);

• to applications, for example in coding theory, cryptography, reverse engineering, algebraic statistics and so on.

ORGANIZERS

Michela Ceria

Department of Computer Science - Università degli Studi di Milano - michela.ceria@gmail.com

André Leroy

Faculté Jean Perrin à Lens - Université d’Artois - andre.leroy@univ-artois.fr

Teo Mora

Department of Mathematics - University of Genoa - 5919@unige.it