Automorphisms mapping a point into a subvariety
Author:
Bjorn Poonen; with an appendix by Matthias Aschenbrenner
Journal:
J. Algebraic Geom. 20 (2011), 785-794
DOI:
https://doi.org/10.1090/S1056-3911-2011-00543-2
Published electronically:
March 14, 2011
MathSciNet review:
2819676
Full-text PDF
Abstract |
References |
Additional Information
Abstract: The problem of deciding, given a complex variety $X$, a point $x \in X$, and a subvariety $Z \subseteq X$, whether there is an automorphism of $X$ mapping $x$ into $Z$ is proved undecidable. Along the way, we prove the undecidability of a version of Hilbert’s tenth problem for systems of polynomials over $\mathbb {Z}$ defining an affine $\mathbb {Q}$-variety whose projective closure is smooth.
References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608
- Matthias Aschenbrenner, Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17 (2004), no. 2, 407–441. MR 2051617, DOI https://doi.org/10.1090/S0894-0347-04-00451-5
- Matthias Aschenbrenner, Uniform degree bounds for Gröbner bases, 2009. In preparation.
- Christine W. Ayoub, On constructing bases for ideals in polynomial rings over the integers, J. Number Theory 17 (1983), no. 2, 204–225. MR 716943, DOI https://doi.org/10.1016/0022-314X%2883%2990021-5
- Edward Bierstone and Pierre D. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11–30. MR 1106412, DOI https://doi.org/10.1007/978-1-4612-0441-1_2
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, DOI https://doi.org/10.1007/s002220050141
- Gábor Bodnár and Josef Schicho, Automated resolution of singularities for hypersurfaces, J. Symbolic Comput. 30 (2000), no. 4, 401–428. MR 1784750, DOI https://doi.org/10.1006/jsco.1999.0414
- A. Clivio, Algorithmic aspects of ${\bf Z}[x_1,\cdots ,x_n]$ with applications to tiling problems, Z. Math. Logik Grundlag. Math. 36 (1990), no. 6, 493–515. MR 1114102, DOI https://doi.org/10.1002/malq.19900360603
- J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425–436. MR 133227, DOI https://doi.org/10.2307/1970289
- Kirsten Eisenträger, Hilbert’s tenth problem for function fields of varieties over $\Bbb C$, Int. Math. Res. Not. 59 (2004), 3191–3205. MR 2097039, DOI https://doi.org/10.1155/S1073792804141354
- Giovanni Gallo and Bhubaneswar Mishra, A solution to Kronecker’s problem, Appl. Algebra Engrg. Comm. Comput. 5 (1994), no. 6, 343–370. MR 1302282, DOI https://doi.org/10.1007/BF01188747
- Patrizia Gianni, Barry Trager, and Gail Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput. 6 (1988), no. 2-3, 149–167. Computational aspects of commutative algebra. MR 988410, DOI https://doi.org/10.1016/S0747-7171%2888%2980040-3
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, DOI https://doi.org/10.1007/BF01206635
- K. H. Kim and F. W. Roush, Diophantine undecidability of ${\bf C}(t_1,t_2)$, J. Algebra 150 (1992), no. 1, 35–44. MR 1174886, DOI https://doi.org/10.1016/S0021-8693%2805%2980047-X
- Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (Russian). MR 0258744
- Wolmer V. Vasconcelos, Flatness testing and torsionfree morphisms, J. Pure Appl. Algebra 122 (1997), no. 3, 313–321. MR 1481094, DOI https://doi.org/10.1016/S0022-4049%2897%2900062-5
- Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852, DOI https://doi.org/10.24033/asens.1573
- O. E. Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629–677. MR 1198092
References
- William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, vol. 3, American Mathematical Society, Providence, RI, 1994. MR 1287608 (95g:13025)
- Matthias Aschenbrenner, Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17 (2004), no. 2, 407–441 (electronic). MR 2051617 (2005c:13032)
- Matthias Aschenbrenner, Uniform degree bounds for Gröbner bases, 2009. In preparation.
- Christine W. Ayoub, On constructing bases for ideals in polynomial rings over the integers, J. Number Theory 17 (1983), no. 2, 204–225. MR 716943 (85m:13017)
- Edward Bierstone and Pierre D. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11–30. MR 1106412 (92h:32053)
- Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306 (98e:14010)
- Gábor Bodnár and Josef Schicho, Automated resolution of singularities for hypersurfaces, J. Symbolic Comput. 30 (2000), no. 4, 401–428. MR 1784750 (2001i:14083)
- A. Clivio, Algorithmic aspects of $\textbf {Z}[x_1,\cdots ,x_n]$ with applications to tiling problems, Z. Math. Logik Grundlag. Math. 36 (1990), no. 6, 493–515. MR 1114102 (92j:13024)
- J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193 (99e:11068)
- Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425–436. MR 0133227 (24:A3061)
- Kirsten Eisenträger, Hilbert’s tenth problem for function fields of varieties over $\mathbb C$, Int. Math. Res. Not. 59 (2004), 3191–3205. MR 2097039 (2005h:11273)
- Giovanni Gallo and Bhubaneswar Mishra, A solution to Kronecker’s problem, Appl. Algebra Engrg. Comm. Comput. 5 (1994), no. 6, 343–370. MR 1302282 (95i:13026)
- Patrizia Gianni, Barry Trager, and Gail Zacharias, Gröbner bases and primary decomposition of polynomial ideals, J. Symbolic Comput. 6 (1988), no. 2-3, 149–167. Computational aspects of commutative algebra. MR 0988410 (90f:68091)
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57:3116)
- Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302
- K. H. Kim and F. W. Roush, Diophantine undecidability of ${\mathbb C}(t_ 1,t_ 2)$, J. Algebra 150 (1992), no. 1, 35–44. MR 1174886 (93h:03062)
- Yu. Matiyasevich, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (Russian). MR 0258744 (41:3390)
- Wolmer V. Vasconcelos, Flatness testing and torsionfree morphisms, J. Pure Appl. Algebra 122 (1997), no. 3, 313–321. MR 1481094 (98i:13013)
- Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852 (90b:14014)
- O. E. Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 6, 629–677. MR 1198092 (93m:14012)
Additional Information
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
MR Author ID:
250625
ORCID:
0000-0002-8593-2792
Email:
poonen@math.mit.edu
Matthias Aschenbrenner
Affiliation:
Department of Mathematics, University of California Los Angeles, P. O. Box 951555, Los Angeles, California 90095-1555
Received by editor(s):
March 27, 2009
Received by editor(s) in revised form:
July 14, 2009
Published electronically:
March 14, 2011
Additional Notes:
The first author was partially supported by NSF grant DMS-0841321. The second author was partially supported by NSF grant DMS-0556197
