A topological approach to undefinability in algebraic extensions of the rationals

Abstract: In 1970 Matiyasevich proved that Hilbert’s Tenth Problem over the integers is undecidable, building on work by Davis-Putnam-Robinson. Hilbert’s Tenth Problem over the rationals is still open, but it could be resolved by giving an existential definition of the integers inside the rationals.

Proving whether such a definition exists is still out of reach. However, we will show that only “very few” algebraic extensions of the rationals have the property that their ring of integers are existentially or universally definable. Equipping the set of all algebraic extensions of the rationals with a natural topology, we show that only a meager subset has this property. An important tool is a new normal form theorem for existential definitions in such extensions. As a corollary, we construct countably many distinct computable algebraic extensions whose rings of integers are neither existentially nor universally definable. Joint work with Russell Miller, Caleb Springer, and Linda Westrick.

logicnumber theory

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic