A universal first-order formula for the ring of integers inside a number field

Jennifer Park studies definability in number fields in connection with Hilbert’s tenth problem over the rationals. An existential definition of the integers inside the rationals, or of the ring of integers inside a number field, would connect the corresponding rational decision problem to the classical undecidability theorem over the integers. The talk instead constructs a definition of a number field’s ring of integers using a single universal quantifier. This extends Koenigsmann’s universal definition of the integers inside the rationals and gives a particularly simple first-order description.