Zahidi, Karim Hilbert’s tenth problem for rings of rational functions. (English) Zbl 1062.03019 Notre Dame J. Formal Logic 43, No. 3, 181-192 (2002). Let \(R\) be a non-constant regular semilocal subring of a field of rational functions \(F(t)\) over an algebraically closed field \(F\) of characteristic \(0\). It is proved that the analogue of Hilbert’s tenth problem for \(R\) (does there exist an algorithm which determines the solvability in \(R\) of systems of multivariate polynomial equations with coefficients in \(\mathbb Z[t]\)?) has a negative answer. Reviewer: Thanases Pheidas (Iraklion) Cited in 3 Documents MSC: 03B25 Decidability of theories and sets of sentences 11U05 Decidability (number-theoretic aspects) 12L05 Decidability and field theory Keywords:decidability; Hilbert’s tenth problem PDFBibTeX XMLCite \textit{K. Zahidi}, Notre Dame J. Formal Logic 43, No. 3, 181--192 (2002; Zbl 1062.03019) Full Text: DOI References: [1] Denef, J., ”The Diophantine problem for polynomial rings and fields of rational functions”, Transactions of the American Mathematical Society , vol. 242 (1978), pp. 391–99. · Zbl 0399.10048 · doi:10.2307/1997746 [2] Kim, K. H., and F. W. Roush, ”Diophantine undecidability of \(\mathbf C(t_ 1,t_ 2)\)”, Journal of Algebra , vol. 150 (1992), pp. 35–44. · Zbl 0754.11039 · doi:10.1016/S0021-8693(05)80047-X [3] Kim, K. H., and F. W. Roush, ”Diophantine unsolvability for function fields over certain infinite fields of characteristic \(p\)”, Journal of Algebra , vol. 152 (1992), pp. 230–39. · Zbl 0768.12008 · doi:10.1016/0021-8693(92)90097-6 [4] Kim, K. H., and F. W. Roush, ”Diophantine unsolvability over \(p\)”-adic function fields, Journal of Algebra , vol. 176 (1995), pp. 83–110. · Zbl 0858.12006 · doi:10.1006/jabr.1995.1234 [5] Matijasevich, Y., ”Enumerable sets are Diophantine”, Doklady Akademii Nauka SSSR , vol. 191 (1970), pp. 272–82. · Zbl 0212.33401 [6] Matsumura, H., Commutative Algebra , W. A. Benjamin, Inc., New York, 1970. · Zbl 0211.06501 [7] Pheidas, T., ”Hilbert’s Tenth Problem for fields of rational functions over finite fields”, Inventiones Mathematicae , vol. 103 (1991), pp. 1–8. · Zbl 0696.12022 · doi:10.1007/BF01239506 [8] Pheidas, T., and K. Zahidi, ”Undecidable existential theories of polynomial rings and function fields”, Communications in Algebra , vol. 27 (1999), pp. 4993–5010. · Zbl 0934.03014 · doi:10.1080/00927879908826744 [9] Poonen, B., ”Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of \(\mathbbQ\)”, preprint, 2003 · Zbl 1028.11077 · doi:10.1090/S0894-0347-03-00433-8 [10] Robinson, J., ”Definability and decision problems in arithmetic”, The Journal of Symbolic Logic , vol. 14 (1949), pp. 98–114. JSTOR: · Zbl 0034.00801 · doi:10.2307/2266510 [11] Robinson, J., ”The undecidability of algebraic rings and fields”, Proceedings of the American Mathematical Society , vol. 10 (1959), pp. 950–57. · Zbl 0100.01501 · doi:10.2307/2033628 [12] Rumely, R. S., ”Undecidability and definability for the theory of global fields”, Transactions of the American Mathematical Society , vol. 262 (1980), pp. 195–217. · Zbl 0472.03010 · doi:10.2307/1999979 [13] Shlapentokh, A., ”Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator”, Inventiones Mathematicae , vol. 129 (1997), pp. 489–507. · Zbl 0887.11053 · doi:10.1007/s002220050170 [14] Shlapentokh, A., ”Diophantine definability over holomorphy rings of algebraic function fields with infinite number of primes allowed as poles”, International Journal of Mathematics , vol. 9 (1998), pp. 1041–66. · Zbl 0920.11080 · doi:10.1142/S0129167X98000440 [15] Silverman, J. H., The Arithmetic of Elliptic Curves , vol. 106 of Graduate Texts in Mathematics , Springer-Verlag, New York, 1986. · Zbl 0585.14026 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.