youtor.org
Russel G Miller Hilberts Tenth Problem for Subrings of the Rational Numbers
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=IBXCg9HsYhM
Hilbert's Tenth Problem for Subrings of the Rational Numbers • Russell Miller • (Queens College, City University of New York (CUNY), USA) • This talk has been given in the KGRC research seminar at the Kurt Gödel Research Centre, Universität Wien on November 15, 2018. • The slides for this talk are available at https://drive.google.com/file/d/131ci... • Abstract • When considering subrings of the field $\\mathbb Q$ of rational numbers, one can view Hilbert's Tenth Problem as an operator, mapping each set $W$ of prime numbers to the set $HTP(R_W)$ of polynomials in $\\mathbb Z[X_1,X_2,\\ldots]$ with solutions in the ring $R_W=\\mathbb Z[W^{-1}]$. The set $HTP(R_{\\emptyset})$ is the original Hilberts Tenth Problem, known since 1970 to be undecidable. If $W$ contains all primes, then one gets $HTP(\\mathbb Q)$, whose decidability status is open. In between lie the continuum-many other subrings of $\\mathbb Q$. • We will begin by discussing topological and measure-theoretic results on the space of all subrings of $\\mathbb Q$, which is homeomorphic to Cantor space. Then we will present a recent result by Ken Kramer and the speaker, showing that the HTP operator does not preserve Turing reducibility. Indeed, in some cases it reverses it: one can have $V lt;_T W$, yet $HTP(R_W) lt;_T HTP(R_V)$. Related techniques reveal that every Turing degree contains a set $W$ which is \\em{HTP-complete}, with $W\\leq_1 HTP(R_W)$. On the other hand, the earlerer results imply that very few sets $W$ have this property: the collection of all HTP-complete sets is meager and has measure $0$ in Cantor space.
#############################
