Geodesic
Representation theorems for r.e. sets and a conjecture related to Poonen's larges subring of $\mathbb Q$
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 50-54 
M. Davis. Representation theorems for r.e. sets and a conjecture related to Poonen's larges subring of $\mathbb Q$. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 50-54. http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a7/
@article{ZNSL_2010_377_a7,
     author = {M. Davis},
     title = {Representation theorems for r.e. sets and a~conjecture related to {Poonen's} larges subring of~$\mathbb Q$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {50--54},
     year = {2010},
     volume = {377},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a7/}
}
TY  - JOUR
AU  - M. Davis
TI  - Representation theorems for r.e. sets and a conjecture related to Poonen's larges subring of $\mathbb Q$
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2010
SP  - 50
EP  - 54
VL  - 377
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a7/
LA  - en
ID  - ZNSL_2010_377_a7
ER  - 
%0 Journal Article
%A M. Davis
%T Representation theorems for r.e. sets and a conjecture related to Poonen's larges subring of $\mathbb Q$
%J Zapiski Nauchnykh Seminarov POMI
%D 2010
%P 50-54
%V 377
%U http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a7/
%G en
%F ZNSL_2010_377_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is remarked that unsolvability results can often be extended to yield novel “representation” theorems for the set of all recursively enumerable sets. In particular it is shown that analysis of the proof of the unsolvability of Hilbert's 10th Problem over Poonen's large subring of $\mathbb Q$ can provide such a theorem. Moreover, applying that theorem to the case of a simple set leads to a conjecture whose truth would imply the unsolvability of Hilbert's 10th Problem over $\mathbb Q$. Bibl. 7 titles.

[1] W. Boone, “The Word Problem”, Ann. Math., 70 (1959), 207–265 | DOI | MR | Zbl

[2] G. Higman, “Subgroups of finitely presented groups”, Proc. Royal Soc. London Series A, 262 (1961), 455–475 | DOI | MR | Zbl

[3] Hilbert's Tenth Problem, MIT Press, Cambridge, Massachusetts, 1993 | MR | Zbl | Zbl

[4] Le dixième problème de Hilbert, Masson, 1995 | MR | Zbl | Zbl

[5] “On the algorithmic insolvability of the word problem in group theory”, American Mathematical Society Translations Ser. 2, 91 (1958), 1–122 | MR | MR | Zbl

[6] B. Poonen, “Hilbert's tenth problem and Mazur's conjecture for large subrings of $\mathbb Q$”, J. Amer. Math. Soc., 16:4 (2003), 981–990 | DOI | MR | Zbl

[7] E. L. Post, “Recursively enumerable sets of positive integers and their decision problems”, Bull. Amer. Math. Soc., 50 (1944), 284–316 ; Reprinted: Martin Davis (ed.), The Undecidable, Raven Press, 1965, and Dover, 2004, 305–337 ; Reprinted: Martin Davis (ed.), Solvability, Provability, Definability, The Collected Works of Emil L. Post, Birkhäuser, 1994, 461–493 | DOI | MR | Zbl | MR | MR

[8] J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag, Berlin–New York, 1994 | MR | Zbl