About: An enumeration operator maps each set A of natural numbers to a set [Formula: see text], in such a way that E(A) can be enumerated uniformly from every enumeration of A. The maximum possible Turing degree of E(A) is therefore the degree of the jump [Formula: see text]. It is impossible to have [Formula: see text] for all A, but possible to achieve this for all A outside a meager set of Lebesgue measure 0. We consider the properties of two specific enumeration operators: the HTP operator, mapping a set W of prime numbers to the set of polynomials realizing Hilbert’s Tenth Problem in the ring [Formula: see text]; and the root operator, mapping the atomic diagram of an algebraic field F of characteristic 0 to the set of polynomials in [Formula: see text] with roots in F. These lead to new open questions about enumeration operators in general.

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About: An enumeration operator maps each set A of natural numbers to a set [Formula: see text], in such a way that E(A) can be enumerated uniformly from every enumeration of A. The maximum possible Turing degree of E(A) is therefore the degree of the jump [Formula: see text]. It is impossible to have [Formula: see text] for all A, but possible to achieve this for all A outside a meager set of Lebesgue measure 0. We consider the properties of two specific enumeration operators: the HTP operator, mapping a set W of prime numbers to the set of polynomials realizing Hilbert’s Tenth Problem in the ring [Formula: see text]; and the root operator, mapping the atomic diagram of an algebraic field F of characteristic 0 to the set of polynomials in [Formula: see text] with roots in F. These lead to new open questions about enumeration operators in general. Goto Sponge NotDistinct Permalink

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type	abstract
value	An enumeration operator maps each set A of natural numbers to a set [Formula: see text], in such a way that E(A) can be enumerated uniformly from every enumeration of A. The maximum possible Turing degree of E(A) is therefore the degree of the jump [Formula: see text]. It is impossible to have [Formula: see text] for all A, but possible to achieve this for all A outside a meager set of Lebesgue measure 0. We consider the properties of two specific enumeration operators: the HTP operator, mapping a set W of prime numbers to the set of polynomials realizing Hilbert’s Tenth Problem in the ring [Formula: see text]; and the root operator, mapping the atomic diagram of an algebraic field F of characteristic 0 to the set of polynomials in [Formula: see text] with roots in F. These lead to new open questions about enumeration operators in general.
subject	Measures (measure theory) Integers Cardinal numbers Diophantine equations Computability theory »more»
part of	Non-coding Enumeration Operators
is abstract of	Non-coding Enumeration Operators
is hasSource of	covid:ann/target/c6d07af4f0b38a2af34d5d6892955341ecc28781 covid:ann/target/2dcb57efcbfe7e9a62e8413a376727d638098114 covid:ann/target/cde8e96d59d03eba9a5dd93cb4aa46321e895570 covid:ann/target/81d800e496c09f9645fe5642f3b3285c0848d72f covid:ann/target/ec200fc6f7f8dba970ad300d4988735566d2a5c1 »more»

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