The theory of the exponential differential equations of semiabelian varieties

By Jonathan Kirby. 2007

70 citations recorded by the source corpus.

The complete first-order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arise from an amalgamation-with-predimension construction in the style of Hrushovski. The theories include necessary and sufficient conditions for a system of equations to have a solution. The necessary conditions generalize Ax’s differential fields version of Schanuel’s conjecture to semiabelian varieties. There is a purely algebraic corollary, the “Weak CIT” for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dafa676eeb2d5b356c65d70f1c02a8d64b924981

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Citations in the corpus

O-minimality and certain atypical intersections (2014, background): “As already remarked, WCA holds for (semi-)abelian varieties, by Ax [3] (see also [23]).”
Exponential algebraicity in exponential fields (2008, methodology): “In particular, they should work for fields with formal analogues of the Weierstrass ℘-functions, and the exponential maps of other semiabelian varieties, using the analogues of Ax’s theorem given in [3].”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, background): “An AxSchanuel result for abelian and indeed semi-abelian varieties is established in work of Ax [6] and Kirby [45], (see also [9, 10]), which thus includes all of the results discussed above except the one concerning the j-function.”