# Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

• [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.)
• Volume: 59, Issue: 5, page 2103-2118
• ISSN: 0373-0956

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## Abstract

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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 ${x}_{1},...,{x}_{n}\in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $ℚ\left({x}_{1},...,{x}_{n}\right)$ has solutions in $R$.

## How to cite

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Moret-Bailly, Laurent, and Shlapentokh, Alexandra. "Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0." Annales de l’institut Fourier 59.5 (2009): 2103-2118. <http://eudml.org/doc/10448>.

@article{Moret2009,
abstract = {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 $x_1,\ldots ,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb\{Q\}(x_1,\ldots ,x_n)$ has solutions in $R$.},
affiliation = {IRMAR Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex (France); East Carolina University Department of Mathematics Greenville, NC 27858 (U.S.A.)},
author = {Moret-Bailly, Laurent, Shlapentokh, Alexandra},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert’s tenth problem; elliptic curves; Diophantine undecidability; Hilbert's tenth problem; undecidability},
language = {eng},
number = {5},
pages = {2103-2118},
publisher = {Association des Annales de l’institut Fourier},
title = {Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0},
url = {http://eudml.org/doc/10448},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 2103
EP - 2118
AB - 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 $x_1,\ldots ,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\mathbb{Q}(x_1,\ldots ,x_n)$ has solutions in $R$.
LA - eng
KW - Hilbert’s tenth problem; elliptic curves; Diophantine undecidability; Hilbert's tenth problem; undecidability
UR - http://eudml.org/doc/10448
ER -

## References

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1. J.-L. Colliot-Thélène, A. Skorobogatov, P. Swinnerton-Dyer, Double fibres and double covers: Paucity of rational points, Acta Arithmetica 79 (1997), 113-135 Zbl0863.14011MR1438597
2. G. Cornelissen, T. Pheidas, K. Zahidi, Division-ample sets and diophantine problem for rings of integers, Journal de Théorie des Nombres Bordeaux 17 (2005), 727-735 Zbl1161.11323MR2212121
3. G. Cornelissen, K. Zahidi, 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 (2000), 253-260, American Mathematical Society Zbl0982.14014MR1802007
4. M. Davis, Hilbert’s tenth problem is unsolvable, American Mathematical Monthly 80 (1973), 233-269 Zbl0277.02008MR317916
5. M. Davis, Y. Matiyasevich, J. Robinson, Hilbert’s tenth problem. Diophantine equations: Positive aspects of a negative solution, Proc. Sympos. Pure Math. 28 (1976), 323-378 Zbl0346.02026MR432534
6. J. Denef, Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220 Zbl0324.02032MR360513
7. J. Denef, The diophantine problem for polynomial rings and fields of rational functions, Transactions of American Mathematical Society 242 (1978), 391-399 Zbl0399.10048MR491583
8. J. Denef, The diophantine problem for polynomial rings of positive characteristic, (1979), 131-145, In M. Boffa, D. van Dalen, and K. MacAloon, editors, North Holland Zbl0457.12011MR567668
9. J. Denef, Diophantine sets of algebraic integers, II, Transactions of American Mathematical Society 257 (1980), 227-236 Zbl0426.12009MR549163
10. J. Denef, L. Lipshitz, Diophantine sets over some rings of algebraic integers, Journal of London Mathematical Society 18 (1978), 385-391 Zbl0399.10049MR518221
11. J. Denef, L. Lipshitz, T. Pheidas, editors, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry, 270 (2000), American Mathematical Society, Providence, RI Zbl0955.00034MR1802007
12. K. Eisenträger, Hilbert’s tenth problem for algebraic function fields of characteristic 2, Pacific J. Math. 210 (2003), 261-281 Zbl1057.11067
13. K. Eisenträger, Hilbert’s tenth problem for function fields of varieties over $ℂ$, Int. Math. Res. Not. (2004), 3191-3205 Zbl1109.11061MR2097039
14. K. Eisenträger, Hilbert’s Tenth Problem for function fields of varieties over number fields and p-adic fields, Journal of Algebra 310 (2007), 775-792 Zbl1152.11050MR2308179
15. M. D. Fried, M. Jarden, Field arithmetic, 11 (2005), Springer Verlag, Berlin, second edition Zbl1055.12003MR2102046
16. H. K. Kim, F. W. Roush, Diophantine undecidability of $ℂ\left({t}_{1},{t}_{2}\right)$, Journal of Algebra 150 (1992), 35-44 Zbl0754.11039MR1174886
17. H. K. Kim, F. W. Roush, Diophantine unsolvability over p-adic function fields, Journal of Algebra 176 (1995), 83-110 Zbl0858.12006MR1345295
18. J. Koenigsmann, Defining transcendentals in function fields, J. Symbolic Logic 67 (2002), 947-956 Zbl1015.03041MR1925951
19. S. Lang, Algebraic Number Theory, (1970), Addison Wesley, Reading, MA Zbl0211.38404MR282947
20. Y. Matiyasevich, Hilbert’s Tenth Problem, (1993), The MIT Press, Cambridge, Massachusetts Zbl0790.03008MR1244324
21. B. Mazur, The topology of rational points, Experimental Mathematics 1 (1992), 35-45 Zbl0784.14012MR1181085
22. B. Mazur, Questions of decidability and undecidability in number theory, Journal of Symbolic Logic 59 (1994), 353-371 Zbl0814.11059MR1276620
23. B. Mazur, Speculation about the topology of rational points: An up-date, Asterisque 228 (1995), 165-181 Zbl0851.14009MR1330932
24. B. Mazur, Open problems regarding rational points on curves and varieties, (1998), Cambridge University Press Zbl0943.14009MR1696485
25. L. Moret-Bailly, Elliptic curves and Hilbert’s Tenth Problem for algebraic function fields over real and $p$-adic fields, Journal für die reine und angewandte Mathematik 587 (2006), 77-143 Zbl1085.14029MR2186976
26. T. Pheidas, Hilbert’s tenth problem for a class of rings of algebraic integers, Proceedings of American Mathematical Society 104 (1988), 611-620 Zbl0697.12020MR962837
27. T. Pheidas, Hilbert’s tenth problem for fields of rational functions over finite fields, Inventiones Mathematicae 103 (1991), 1-8 Zbl0696.12022MR1079837
28. T. Pheidas, Endomorphisms of elliptic curves and undecidability in function fields of positive characteristic, J. Algebra 273 (2004), 395-411 Zbl1035.11064MR2032468
29. B. Poonen, Hilbert’s Tenth Problem and Mazur’s conjecture for large subrings of $ℚ$, Journal of AMS 16 (2003), 981-990 Zbl1028.11077MR1992832
30. B. Poonen, A. Shlapentokh, 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), 27-48 Zbl1139.11056MR2196727
31. Florian Pop, Elementary equivalence versus isomorphism, Invent. Math. 150 (2002), 385-408 Zbl1162.12302MR1933588
32. H. Shapiro, A. Shlapentokh, Diophantine relations between algebraic number fields, Communications on Pure and Applied Mathematics XLII (1989), 1113-1122 Zbl0698.12022MR1029120
33. A. Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number fields, Communications on Pure and Applied Mathematics XLII (1989), 939-962 Zbl0695.12020MR1008797
34. A. Shlapentokh, Hilbert’s tenth problem for rings of algebraic functions of characteristic $0$, J. Number Theory 40 (1992), 218-236 Zbl0746.03008MR1149739
35. A. Shlapentokh, Diophantine classes of holomorphy rings of global fields, Journal of Algebra 169 (1994), 39-175 Zbl0810.11073MR1296586
36. A. Shlapentokh, Diophantine undecidability for some holomorphy rings of algebraic functions of characteristic 0, Communications in Algebra 22 (1994), 4379-4404 Zbl0810.11074MR1284336
37. A. Shlapentokh, Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of $ℚ$, Annals of Pure and Applied Logic 68 (1994), 299-325 Zbl0816.11066MR1289287
38. A. Shlapentokh, Diophantine undecidability of algebraic function fields over finite fields of constants, Journal of Number Theory 58 (1996), 317-342 Zbl0856.11058MR1393619
39. A. Shlapentokh, Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Inventiones Mathematicae 129 (1997), 489-507 Zbl0887.11053MR1465332
40. A. Shlapentokh, Diophantine undecidability of function fields of characteristic greater than 2 finitely generated over a field algebraic over a finite field, Compositio Mathematica 132 (2002), 99-120 Zbl1011.03026MR1914257
41. A. Shlapentokh, On diophantine decidability and definability in some rings of algebraic functions of characteristic 0, Journal of Symbolic Logic 67 (2002), 759-786 Zbl1011.03027MR1905166
42. A. Shlapentokh, 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 95 (2002), 227-252 Zbl1082.11079MR1924099
43. A. Shlapentokh, A ring version of Mazur’s conjecture on topology of rational points, International Mathematics Research Notices 7 (2003), 411-423 Zbl1107.11049MR1939572
44. A. Shlapentokh, On diophantine definability and decidability in some infinite totally real extensions of $ℚ$, Transactions of AMS 356 (2004), 3189-3207 Zbl1052.11082MR2052946
45. A. Shlapentokh, First-order definitions of rational functions and $𝒮$-integers over holomorphy rings of algebraic functions of characteristic 0, Ann. Pure Appl. Logic 136 (2005), 267-283 Zbl1079.03025MR2169686
46. A. Shlapentokh, Hilbert’s Tenth Problem: Diophantine Classes and Extensions to Global Fields, (2006), Cambridge University Press Zbl1196.11166MR2297245
47. A. Shlyapentokh, Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 304 (2003), 141-167, 171 Zbl1140.11356MR2054753
48. C. Videla, Hilbert’s tenth problem for rational function fields in characteristic 2, Proceedings of the American Mathematical Society 120 (1994), 249-253 Zbl0795.03015MR1159179
49. K. Zahidi, The existential theory of real hyperelliptic fields, Journal of Algebra 233 (2000), 65-86 Zbl0985.11062MR1793590
50. K. Zahidi, Hilbert’s tenth problem for rings of rational functions, Notre Dame Journal of Formal Logic 43 (2003), 181-192 Zbl1062.03019MR2034745

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