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Preprint Number 1761

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1761. Jan Dobrowolski
Sets, groups, and fields definable in vector spaces with a bilinear form

Submission date: 15 April 2020


We study definable sets, groups, and fields in the theory T_∞ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an (ℕ × ℤ,≤_{lex})-valued dimension on definable sets in T_∞ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in T_∞ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in T_∞ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in T_∞, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component.
We also consider the theory T^{RCF}_∞ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of T_∞, we define a dimension on sets definable in T^{RCF}_∞, and using it we prove analogous results about definable groups and fields: every group definable in T^{RCF}_∞ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in T^{RCF}_∞ is definable in the field of scalars, hence it is either real closed or algebraically closed.

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Full text arXiv 2004.07238: pdf, ps.

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