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Classes of (Lie) Differential Fields without Model Companions

Friday 8th April 2005 - 12:00 to 12:30
INI Seminar Room 1

A Lie differential field is a field F given with some Lie algebra L acting on F as derivations. If we fix L we get the class of L-differential fields, which has amalgamation when the characteristic is 0. If in addition L is finite dimensional over F then the above class has a model companion (and hence a model completion).

However if L isn't finitely presented (at least locally), i.e. if there is a finitely generated sub Lie algebra of L without a finite presentation, then the class of L-differential fields does NOT have a model companion. I will describe how this result is proved by producing a non-eliminable quantifier, using a system of linear PDEs. The question of a tighter connection between companionability and finite presentability remains open.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons