Rational places in extensions and sequences of function fields of Kummer type

New examples of asymptotically good Kummer type towers

“…As we said in the Introduction, it is well-known that a tame recursive tower is asymptotically good if it has non-empty splitting locus and finite ramification locus. The next result, proved in [1], gives sufficient conditions in order to have non-empty splitting locus, in the particular class of sequences of Kummer type recursively defined by (1.1).…”

“…and this implies that x i (Q) = γ for some γ ∈ F q such that b 1 (γ) = 0. Thus x i (Q) ∈ S 0 by (1). Suppose now that P i+1 is a pole of x i+1 .…”

“…The aim of this paper is to continue the investigation (initiated in [1]) of the asymptotic behavior of towers of function fields defined by a Kummer equation of the form…”

“…where f ∈ F q [x] is a suitable polynomial and α ∈ F * q . More precisely, in [1] we obtained conditions in order to have non empty splitting locus of towers recursively defined by (1.1), and now we will deal with the ramification locus. The finiteness of the ramification locus of towers recursively defined by (1.1) will suffice to prove their good asymptotic behavior because an important and well-known result of Garcia and Stichtenoth states that if a tame tower has non-empty splitting locus and finite ramification locus, then the tower is asymptotically good, [4, Theorem 2.1].…”