930 research outputs found
The theory of the exponential differential equations of semiabelian varieties
The complete first order theories of the exponential differential equations
of semiabelian varieties are given. It is shown that these theories also arises
from an amalgamation-with-predimension construction in the style of Hrushovski.
The theory includes necessary and sufficient conditions for a system of
equations to have a solution. The necessary condition generalizes 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.Comment: 53 pages; v3: Substantial changes, including a completely new
introductio
On algebraic relations between solutions of a generic Painleve equation
We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve
equation (i.e. an equation in one of the families PI-PVI but with the complex
parameters \alpha, \beta,.. algebraically independent) then any algebraic
dependence over C(t) between a set of solutions and their derivatives
(y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their
derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school
on "irreducibility" of the Painleve equations, with the trichomoty theorem for
strongly minimal sets in differentially closed fields.Comment: 23 page
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
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