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Pseudoexponential fields are exponential fields similar to complex exponentiation which satisfy the Schanuel Property, i.e., the abstract statement of Schanuel’s Conjecture, and an adapted form of existential closure.

Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Tapani Hyttinen , Random logarithm and homogeneity, Logic and Its Applications ( Andreas Blass and Yi Zhang , editors), Contemporary Mathematics, pp. 137166, American Mathematical Society, Providence, RI, 2005.

Jonathan Kirby , A Note on the Axioms for Zilber’s Pseudo-Exponential Fields. Notre Dame Journal of Formal Logic, vol. 54, no. 3–4, pp. 509520, 2013.

Jonathan Kirby , Finitely presented exponential fields. Algebra & Number Theory, vol. 7, no. 4, pp. 943980, 2013.

David W. Masser , Specialisations of finitely generated subgroups of abelian varieties. Transactions of the American Mathematical Society, vol. 311, no. 1, pp. 413424, 1989.

Boris Zilber , Pseudo-exponentiation on algebraically closed fields of characteristic zero. Annals of Pure and Applied Logic, vol. 132, no. 1, pp. 6795, 2005.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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