Ax-Schanuel for Linear Differential Equations

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 J. Kirby and C. Crampin 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.

24 pages. Minor changes in Sections 1-5. Section 6 has been significantly modified

References in corpus (2)

• The theory of the exponential differential equations of semiabelian varieties
• Definability of derivations in the reducts of differentially closed fields

Cited by in corpus (2)

• Ax-Schanuel and strong minimality for the $j$-function
• Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields