Linear Algebra, Old and New

The purpose of my talk is to present old and new results which lie on the border between Linear Algebra and Computational Commutative Algebra. The main source is the recent book Computational Linear and Commutative Algebra by Martin Kreuzer and myself. The talk starts by recalling some old results related to one endomorphism of a finitely generated vector space. Then we progress to considering families of pairwise commuting endomorphisms: this opens up a new world. The merits of this modern approach to linear algebra will be described. Among them, we will see the connection to many fundamental problems in computer algebra such as computing the primary decomposition of zero-dimensional ideals, and solving systems of polynomial equations. The final part of the talk will be devoted to exhibiting a link between advanced tools in linear algebra and an old theme in algebraic geometry.

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