Abstract
We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.
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This work was supported by the University of Oxford Dulverton Scholarship.
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Aslanyan, V. Ax–Schanuel for linear differential equations. Arch. Math. Logic 57, 629–648 (2018). https://doi.org/10.1007/s00153-017-0602-3
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DOI: https://doi.org/10.1007/s00153-017-0602-3
Keywords
- Model-theoretic algebra
- Abstract differential equation
- Ax–Schanuel theorem
- Predimension
- Hrushovski construction