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

Hilbert's tenth problem over the rational numbers Q (or, any number field K) asks the following: given a polynomial in several variables with coefficients in Q (resp. K), is there a general algorithm that decides whether this polynomial has a solution in Q (resp. K)? Unlike the classical Hilbert's tenth problem over Z, this problem is still unresolved. To reduce this problem to the classical problem, we need a definition of Z in Q (resp. ring of integers in K) using only an existential quantifier. This problem is still open. I will present a definition of the ring of integers in a number field, which uses only one universal quantifier, which is, in a sense, the simplest logical description that we can hope for. This is a generalization of Koenigsmann's work, which defines Z in Q using one universal quantifier.

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