Undecidability of parametric solutions of polynomial equations

Authors:
K. H. Kim and F. W. Roush

Journal:
Proc. Amer. Math. Soc. **118** (1993), 345-348

MSC:
Primary 03D35; Secondary 03D80, 11U05, 12L05

DOI:
https://doi.org/10.1090/S0002-9939-1993-1132414-5

MathSciNet review:
1132414

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that, for any field of characteristic 0 satisfying a hypothesis related to not being algebraically closed, the problem of finding non-constant parametric solutions in to a polynomial system with coefficients in is algorithmically unsolvable.

**[BDL]**J. Becker, J. Denef, and L. Lipshitz,*Further remarks on the elementary theory of formal power series rings*, Model Theory of Algebra and Arithmetic (Proc. Karpacz, Poland 1979), Lectures Notes in Math., vol. 834, Springer-Verlag, Berlin, 1980. MR**606776 (83a:13013)****[C]**G. L. Cherlin,*Definability in power series rings of nonzero characteristic*, Models and Sets, Lecture Notes in Math., vol. 1103, Springer-Verlag, Berlin, 1984, pp. 102-112. MR**775690 (86h:03059)****[D1]**J. Denef,*The Diophantine problem for polynomial rings and fields of rational functions*, Trans. Amer. Math. Soc.**242**(1978), 391-399. MR**0491583 (58:10809)****[D2]**-,*The Diophantine problem for polynomial rings of positive characteristic*, Logic Colloq., vol. 82, North-Holland, Amsterdam, 1979. MR**567668 (81h:03090)****[Ha]**R. Hartshorne,*Algebraic geometry*, Springer-Verlag, Berlin, 1977. MR**0463157 (57:3116)****[KR]**K. H. Kim and F. W. Roush,*Diophantine undecidability of*, J. Algebra (to appear).**[P]**T. Pheidas,*An undecidability result for power series rings of positive characteristic*. II, Proc. Amer. Math. Soc.**100**(1987), 526-530. MR**891158 (89c:03075)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1132414-5

Keywords:
Diophantine problem in polynomial equation,
parametric solution

Article copyright:
© Copyright 1993
American Mathematical Society