The bifurcation sets of polynomial functions have been studied by many mathematicians from various points view. In particular, Némethi and Zaharia described them in terms Newton polytopes. this paper, we will show analogous results for rational functions.

We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class Σα (respectively Π 0 α). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a Σ1-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is a...

We prove the following surprising result: there exist a 1-counter Büchi automaton and a 2-tape Büchi automaton such that the ω-language of the first and the infinitary rational relation of the second in one model of ZFC are Π2-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an ω-language accepted by a 1-counter Büchi au...

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-languageL(A) accepted by a Büchi 1-counter automaton A. We prove the following surprising result: there exists a 1-counter Büchi automaton A such that the cardinality of th...