Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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
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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 [Enhancements On Off] (What's this?)

  • [AL94] 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)
  • [Asc04] 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)
  • [Asc09] Matthias Aschenbrenner, Uniform degree bounds for Gröbner bases, 2009. In preparation.
  • [Ayo83] 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)
  • [BM91] 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)
  • [BM97] 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)
  • [BS00] Gábor Bodnár and Josef Schicho, Automated resolution of singularities for hypersurfaces, J. Symbolic Comput. 30 (2000), no. 4, 401-428. MR 1784750 (2001i:14083)
  • [Cli90] 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 (92j:13024)
  • [Cre97] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193 (99e:11068)
  • [DPR61] 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)
  • [Eis04] Kirsten Eisenträger, Hilbert's tenth problem for function fields of varieties over $ \mathbb{C}$, Int. Math. Res. Not. (2004), no. 59, 3191-3205. MR 2097039 (2005h:11273)
  • [GM94] 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)
  • [GTZ88] 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)
  • [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57:3116)
  • [Her26] Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736-788 (German). MR 1512302
  • [KR92] K. H. Kim and F. W. Roush, Diophantine undecidability of $ {\mathbb{C}}(t\sb 1,t\sb 2)$, J. Algebra 150 (1992), no. 1, 35-44. MR 1174886 (93h:03062)
  • [Mat70] Yu. Matiyasevich, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279-282 (Russian). MR 0258744 (41:3390)
  • [Vas97] Wolmer V. Vasconcelos, Flatness testing and torsionfree morphisms, J. Pure Appl. Algebra 122 (1997), no. 3, 313-321. MR 1481094 (98i:13013)
  • [Vil89] Orlando Villamayor, Constructiveness of Hironaka's resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1-32. MR 985852 (90b:14014)
  • [VU92] 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
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

DOI: https://doi.org/10.1090/S1056-3911-2011-00543-2
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

American Mathematical Society