# zbMATH — the first resource for mathematics

Hilbert’s tenth problem for fields of rational functions over finite fields. (English) Zbl 0696.12022
Let F be a finite field of characteristic $$p>2$$ and F(t) the field of rational functions in the variable t with coefficients in F. It is proved that the existential theory of F(t) in the language $$\{+,\cdot,0,1,t\}$$ is undecidable, and so, there is no algorithm to solve arbitrary polynomial equations over F(t); this is an analogue to Hilbert’s tenth problem for the case of a function field. Due to the similarities between $${\mathbb{Q}}$$ (the field of rationals) and the common properties of function fields over finite fields, when it comes to the solvability of diophantine equations, this result seems to suggest that the analogue of Hilbert’s tenth problem for $${\mathbb{Q}}$$ has a negative answer.
Reviewer: T.Pheidas

##### MSC:
 11U05 Decidability (number-theoretic aspects) 11R58 Arithmetic theory of algebraic function fields 11D99 Diophantine equations
Full Text:
##### References:
 [1] Cherlin, G.: Undecidability of rational function fields in nonzero characteristic, Logic colloquium ’82, Lolli, G., Logo, G., Marcja A. (eds), pp. 85-95. North Holland 1984 · Zbl 0551.03027 [2] Davis, M., Matijasevic, Yu., Robinson, J.: Diophantine equations: Positive Aspects of a negative solution. Proc. Symp. Pure Math.28, 323-378 (1976) · Zbl 0346.02026 [3] Denef, J.: The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium ’78, Boffa, M., van Dalen, D., McAloon, K. (eds), pp. 131-145. North Holland 1979 · Zbl 0457.12011 [4] Denef, J.: The Diophantine problem for polynomial rings and fields of rational functions. Trans. Am. Math. Soc.242, 391-399 (1978) · Zbl 0399.10048 · doi:10.1090/S0002-9947-1978-0491583-7 [5] Denef, J.: Diophantine sets over algebraic integer rings II. Trans. Am. Math. Soc.257, 227-336 (1980) · Zbl 0426.12009 · doi:10.1090/S0002-9947-1980-0549163-X [6] Denef, J., Lipshitz, L.: Diophantine sets over some rings of algebraic integers. J. Lond. Math. Soc.18, 385-391 (1978). · Zbl 0399.10049 · doi:10.1112/jlms/s2-18.3.385 [7] Ersov, Yu.: Undecidability of certain fields. Dokl. Akad. Nauk SSSR161, 349-352 (1965) [8] Lang, S.: Algebra, Reading, MA: Addison Wesley 1971 [9] Matijasevic, Yu.: Enumerable sets are Diophantine. Dokl. Akad. Nauk SSSR191, 279-282 (1970) [10] Penzin, Yu.: Undecidability of fields of rational functions over fields of characteristic 2. Algebra Logic12, 205-219 (1973) · Zbl 0282.02019 · doi:10.1007/BF02219294 [11] Pheidas, T.: An undecidability result for power series rings of positive characteristic II. Proc. Am. Math. Soc.100, 526-530 (1987) · Zbl 0664.03008 · doi:10.1090/S0002-9939-1987-0891158-2 [12] Pheidas, T.: Hilbert’s Tenth Problem for a class of rings of algebraic integers. Proc. Am. Math. Soc.104, 611-620 (1988) · Zbl 0697.12020 [13] Robinson, J.: Definability and decision problems in arithmetic. J. Symb. Logic14, 98-114 (1949) · Zbl 0034.00801 · doi:10.2307/2266510 [14] Rumely, R.: Undecidability and definability for the theory of global fields. Trans. Am. Math. Soc.262, 195-217 (1980) · Zbl 0472.03010 · doi:10.1090/S0002-9947-1980-0583852-6
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.