The main goal of this work is to answer a question of Dèbes and Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials.

We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, compute explicitly corresponding rates. These are identified, as is well known, case terms an appropriate Hilbertian norm; entropy, a Dirichlet form which morphs into version familiar Fisher information under conditions detailed balance. Here we obtain trajectorial versions these res...

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...