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

Connections for Women: Model Theory and Its Interactions with Number Theory and Arithmetic Geometry February 10, 2014 - February 11, 2014

February 10, 2014 (11:45 AM PST - 12:30 PM PST)
Speaker(s): Jennifer Park (Massachusetts Institute of Technology)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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.

20174?type=thumb Park notes 147 KB application/pdf Download
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