2010 Summer Course on Model Theory : Chapter 1
نویسنده
چکیده
1) The Nullstellensatz and the R-Nullstellensatz. 2) Chevalley’s Theorem: the image of a constructible set is constructible. 3) (Grothendieck, Ax) An injective polynomial map from C to C is surjective. 4) Hilbert’s 17th problem: a positive semidefinite rational function f ∈ R(t1, . . . , tn) is a sum of squares. 5) Polynomially compact operators have invariant subspaces. (Let V be a complex Hilbert space, L a bounded linear operator on H, and P (t) ∈ C[t] a nonzero polynomial. Suppose that P (L) is a compact operator: the image of the unit ball has compact closure. Then there exists a nontrivial, proper closed subspace W of V which is L-invariant.) 6) (Ax-Kochen) For each fixed n ∈ Z and all sufficiently large primes p, a homogeneous form with coefficients in Qp with at least n + 1 variables has a nontrivial zero. 7) (Duesler-Knecht) An analogue of the Ax-Katz theorem for rationally connected varieties over the maximal unramified extension of Qp. 8) (Faltings, Hrushovski) Mordell-Lang Conjecture.
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تاریخ انتشار 2010