Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
[Indécidabilité diophantienne des anneaux d’holomorphie de corps de fonctions de caractéristique nulle]
Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2103-2118.

Soit K un corps de fonctions d’une variable sur un corps de caractéristique nulle. Soit R un anneau d’holomorphie de K, distinct de K. Si K est récursif, nous démontrons que le dixième problème de Hilbert sur R est indécidable. En général, il existe x 1 ,...,x n dans R tels qu’il n’y ait pas d’algorithme décidant si une équation polynomiale à coefficients dans (x 1 ,...,x n ) a une solution dans R.

Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: if K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x 1 ,...,x n R such that there is no algorithm to tell whether a polynomial equation with coefficients in (x 1 ,...,x n ) has solutions in R.

DOI : 10.5802/aif.2484
Classification : 11U05, 03D35, 11G05
Keywords: Hilbert’s tenth problem, elliptic curves, Diophantine undecidability
Mot clés : dixième problème de Hilbert, courbes elliptiques, indécidabilité diophantienne
Moret-Bailly, Laurent 1 ; Shlapentokh, Alexandra 2

1 IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex (France)
2 East Carolina University Department of Mathematics Greenville, NC 27858 (U.S.A.)
@article{AIF_2009__59_5_2103_0,
     author = {Moret-Bailly, Laurent and Shlapentokh, Alexandra},
     title = {Diophantine {Undecidability} of {Holomorphy} {Rings} of {Function} {Fields} of {Characteristic~0}},
     journal = {Annales de l'Institut Fourier},
     pages = {2103--2118},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {5},
     year = {2009},
     doi = {10.5802/aif.2484},
     zbl = {1226.11131},
     mrnumber = {2573198},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2484/}
}
TY  - JOUR
AU  - Moret-Bailly, Laurent
AU  - Shlapentokh, Alexandra
TI  - Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2103
EP  - 2118
VL  - 59
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2484/
DO  - 10.5802/aif.2484
LA  - en
ID  - AIF_2009__59_5_2103_0
ER  - 
%0 Journal Article
%A Moret-Bailly, Laurent
%A Shlapentokh, Alexandra
%T Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
%J Annales de l'Institut Fourier
%D 2009
%P 2103-2118
%V 59
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2484/
%R 10.5802/aif.2484
%G en
%F AIF_2009__59_5_2103_0
Moret-Bailly, Laurent; Shlapentokh, Alexandra. Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2103-2118. doi : 10.5802/aif.2484. https://aif.centre-mersenne.org/articles/10.5802/aif.2484/

[1] Colliot-Thélène, J.-L.; Skorobogatov, A.; Swinnerton-Dyer, P. Double fibres and double covers: Paucity of rational points, Acta Arithmetica, Volume 79 (1997), pp. 113-135 | MR | Zbl

[2] Cornelissen, G.; Pheidas, T.; Zahidi, K. Division-ample sets and diophantine problem for rings of integers, Journal de Théorie des Nombres Bordeaux, Volume 17 (2005), pp. 727-735 | DOI | Numdam | MR | Zbl

[3] Cornelissen, G.; Zahidi, K. Topology of diophantine sets: Remarks on Mazur’s conjectures, In Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry, volume 270 of Contemporary Mathematics, American Mathematical Society, 2000, pp. 253-260 | MR | Zbl

[4] Davis, M. Hilbert’s tenth problem is unsolvable, American Mathematical Monthly, Volume 80 (1973), pp. 233-269 | DOI | MR | Zbl

[5] Davis, M.; Matiyasevich, Y.; Robinson, J. Hilbert’s tenth problem. Diophantine equations: Positive aspects of a negative solution, Proc. Sympos. Pure Math., Volume 28 (1976), pp. 323- 378 (Amer. Math. Soc.) | MR | Zbl

[6] Denef, J. Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc., Volume 48 (1975), pp. 214-220 | MR | Zbl

[7] Denef, J. The diophantine problem for polynomial rings and fields of rational functions, Transactions of American Mathematical Society, Volume 242 (1978), pp. 391-399 | DOI | MR | Zbl

[8] Denef, J. The diophantine problem for polynomial rings of positive characteristic, Logic Colloquium 78 (1979), pp. 131-145 | MR | Zbl

[9] Denef, J. Diophantine sets of algebraic integers, II, Transactions of American Mathematical Society, Volume 257 (1980) no. 1, pp. 227-236 | DOI | MR | Zbl

[10] Denef, J.; Lipshitz, L. Diophantine sets over some rings of algebraic integers, Journal of London Mathematical Society, Volume 18 (1978) no. 2, pp. 385-391 | DOI | MR | Zbl

[11] Denef, J.; Lipshitz, L.; Pheidas, editors, T. Hilbert’s tenth problem: relations with arithmetic and algebraic geometry, Contemporary Mathematics, 270, American Mathematical Society, Providence, RI, 2000 (Papers from the workshop held at Ghent University, Ghent, November 2-5, 1999 ) | MR | Zbl

[12] Eisenträger, K. Hilbert’s tenth problem for algebraic function fields of characteristic 2, Pacific J. Math., Volume 210 (2003) no. 2, pp. 261-281 | DOI | Zbl

[13] Eisenträger, K. Hilbert’s tenth problem for function fields of varieties over , Int. Math. Res. Not. (2004) no. 59, pp. 3191-3205 | DOI | MR | Zbl

[14] Eisenträger, K. Hilbert’s Tenth Problem for function fields of varieties over number fields and p-adic fields, Journal of Algebra, Volume 310 (2007), pp. 775-792 | DOI | MR | Zbl

[15] Fried, M. D.; Jarden, M. Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 11, Springer Verlag, Berlin, second edition, 2005 | MR | Zbl

[16] Kim, H. K.; Roush, F. W. Diophantine undecidability of (t 1 ,t 2 ), Journal of Algebra, Volume 150 (1992) no. 1, pp. 35-44 | DOI | MR | Zbl

[17] Kim, H. K.; Roush, F. W. Diophantine unsolvability over p-adic function fields, Journal of Algebra, Volume 176 (1995), pp. 83-110 | DOI | MR | Zbl

[18] Koenigsmann, J. Defining transcendentals in function fields, J. Symbolic Logic, Volume 67 (2002) no. 3, pp. 947-956 | DOI | MR | Zbl

[19] Lang, S. Algebraic Number Theory, Addison Wesley, Reading, MA, 1970 | MR | Zbl

[20] Matiyasevich, Y. Hilbert’s Tenth Problem, The MIT Press, Cambridge, Massachusetts, 1993 | MR | Zbl

[21] Mazur, B. The topology of rational points, Experimental Mathematics, Volume 1 (1992) no. 1, pp. 35-45 | MR | Zbl

[22] Mazur, B. Questions of decidability and undecidability in number theory, Journal of Symbolic Logic, Volume 59 (1994) no. 2, pp. 353-371 | DOI | MR | Zbl

[23] Mazur, B. Speculation about the topology of rational points: An up-date, Asterisque, Volume 228 (1995), pp. 165-181 | MR | Zbl

[24] Mazur, B. Open problems regarding rational points on curves and varieties, Galois Representations in Arithmetic Algebraic Geometry, Cambridge University Press, 1998 | MR | Zbl

[25] Moret-Bailly, L. Elliptic curves and Hilbert’s Tenth Problem for algebraic function fields over real and p-adic fields, Journal für die reine und angewandte Mathematik, Volume 587 (2006), pp. 77-143 | MR | Zbl

[26] Pheidas, T. Hilbert’s tenth problem for a class of rings of algebraic integers, Proceedings of American Mathematical Society, Volume 104 (1988) no. 2, pp. 611-620 | MR | Zbl

[27] Pheidas, T. Hilbert’s tenth problem for fields of rational functions over finite fields, Inventiones Mathematicae, Volume 103 (1991), pp. 1-8 | DOI | MR | Zbl

[28] Pheidas, T. Endomorphisms of elliptic curves and undecidability in function fields of positive characteristic, J. Algebra, Volume 273 (2004) no. 1, pp. 395-411 | DOI | MR | Zbl

[29] Poonen, B. Hilbert’s Tenth Problem and Mazur’s conjecture for large subrings of , Journal of AMS, Volume 16 (2003) no. 4, pp. 981-990 | MR | Zbl

[30] Poonen, B.; Shlapentokh, A. Diophantine definability of infinite discrete non-archimedean sets and diophantine models for large subrings of number fields, Journal für die Reine und Angewandte Mathematik (2005), pp. 27-48 | DOI | MR | Zbl

[31] Pop, Florian Elementary equivalence versus isomorphism, Invent. Math., Volume 150 (2002) no. 2, pp. 385-408 | DOI | MR | Zbl

[32] Shapiro, H.; Shlapentokh, A. Diophantine relations between algebraic number fields, Communications on Pure and Applied Mathematics, Volume XLII (1989), pp. 1113-1122 | DOI | MR | Zbl

[33] Shlapentokh, A. Extension of Hilbert’s tenth problem to some algebraic number fields, Communications on Pure and Applied Mathematics, Volume XLII (1989), pp. 939-962 | DOI | MR | Zbl

[34] Shlapentokh, A. Hilbert’s tenth problem for rings of algebraic functions of characteristic 0, J. Number Theory, Volume 40 (1992) no. 2, pp. 218-236 | DOI | MR | Zbl

[35] Shlapentokh, A. Diophantine classes of holomorphy rings of global fields, Journal of Algebra, Volume 169 (1994) no. 1, pp. 39-175 | DOI | MR | Zbl

[36] Shlapentokh, A. Diophantine undecidability for some holomorphy rings of algebraic functions of characteristic 0, Communications in Algebra, Volume 22 (1994) no. 11, pp. 4379-4404 | DOI | MR | Zbl

[37] Shlapentokh, A. Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of , Annals of Pure and Applied Logic, Volume 68 (1994), pp. 299-325 | DOI | MR | Zbl

[38] Shlapentokh, A. Diophantine undecidability of algebraic function fields over finite fields of constants, Journal of Number Theory, Volume 58 (1996) no. 2, pp. 317-342 | DOI | MR | Zbl

[39] Shlapentokh, A. Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Inventiones Mathematicae, Volume 129 (1997), pp. 489-507 | DOI | MR | Zbl

[40] Shlapentokh, A. Diophantine undecidability of function fields of characteristic greater than 2 finitely generated over a field algebraic over a finite field, Compositio Mathematica, Volume 132 (2002) no. 1, pp. 99-120 | DOI | MR | Zbl

[41] Shlapentokh, A. On diophantine decidability and definability in some rings of algebraic functions of characteristic 0, Journal of Symbolic Logic, Volume 67 (2002) no. 2, pp. 759-786 | DOI | MR | Zbl

[42] Shlapentokh, A. On diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2, Journal of Number Theory, Volume 95 (2002), pp. 227-252 | MR | Zbl

[43] Shlapentokh, A. A ring version of Mazur’s conjecture on topology of rational points, International Mathematics Research Notices, Volume 7 (2003), pp. 411-423 | DOI | MR | Zbl

[44] Shlapentokh, A. On diophantine definability and decidability in some infinite totally real extensions of , Transactions of AMS, Volume 356 (2004) no. 8, pp. 3189-3207 | DOI | MR | Zbl

[45] Shlapentokh, A. First-order definitions of rational functions and 𝒮-integers over holomorphy rings of algebraic functions of characteristic 0, Ann. Pure Appl. Logic, Volume 136 (2005) no. 3, pp. 267-283 | DOI | MR | Zbl

[46] Shlapentokh, A. Hilbert’s Tenth Problem: Diophantine Classes and Extensions to Global Fields, Cambridge University Press, 2006 | MR

[47] Shlyapentokh, A. Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 304 (2003) no. Teor. Slozhn. Vychisl. 8, p. 141-167, 171 | MR | Zbl

[48] Videla, C. Hilbert’s tenth problem for rational function fields in characteristic 2, Proceedings of the American Mathematical Society, Volume 120 (1994) no. 1, pp. 249-253 | MR | Zbl

[49] Zahidi, K. The existential theory of real hyperelliptic fields, Journal of Algebra, Volume 233 (2000) no. 1, pp. 65-86 | MR | Zbl

[50] Zahidi, K. Hilbert’s tenth problem for rings of rational functions, Notre Dame Journal of Formal Logic, Volume 43 (2003), pp. 181-192 | DOI | MR | Zbl

Cité par Sources :