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Projective sets and ordinary differential equations

Authors:	Alessandro Andretta Alberto Marcone

Institution:	Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy ; Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

Abstract:	We prove that for $n \geq 2$ the set of Cauchy problems of dimension which have a global solution is $\boldsymbol\Sigma_{1}^{1}$ -complete and that the set of ordinary differential equations which have a global solution for every initial condition is $\boldsymbol\Pi_{1}^{1}$ -complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for $n \geq 2$ the set of Cauchy problems of dimension which have a global solution even if we perturb a bit the initial condition is $\boldsymbol\Pi_{2}^{1}$ -complete.

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