A fundamental theorem is proved for tropical PDEs over valued fields, with the corollary that radii of convergence for ODE solutions can be computed tropically.
Geometry over the tropical dual numbers
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abstract
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach in classical algebraic geometry. The bend loci of an ideal over tropical dual numbers coincides with the bend loci of a congruence over tropical semiring and this enables tropical dual numbers to serve as an algebraic structure for tropical geometry. Tropical Zariski topology on affine tropical spaces whose closed sets are non-linear loci of ideals over tropical dual numbers offers an alternative point of view to strong Zariski topology defined by Giansiracusa.
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The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations
A fundamental theorem is proved for tropical PDEs over valued fields, with the corollary that radii of convergence for ODE solutions can be computed tropically.