By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)
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Extra resources for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings
Alexandra Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number ﬁelds, Comm. Pure Appl. Math. 42 (1989), no. 7, 939–962. Shl92. Alexandra Shlapentokh, Hilbert’s tenth problem for rings of algebraic functions in one variable over ﬁelds of constants of positive characteristic, Trans. Amer. Math. Soc. 333 (1992), no. 1, 275–298. Shl00a. Alexandra Shlapentokh, Hilbert’s tenth problem for algebraic function ﬁelds over inﬁnite ﬁelds of constants of positive characteristic, Paciﬁc J.
Let D ∈ OF denote the discriminant of this basis. If I is an ideal in OK , let NK/Q (I) ∈ Z≥0 denote its norm. Hilbert’s Tenth Problem over Rings of Algebraic Integers 37 Lemma 4. There is a positive integer c > 0 depending only on F , K, and α such that the following holds. Let I ⊂ OK be a nonzero nonunit ideal, and let s−1 μ ∈ OK . Write μ = i=0 ai αi with ai ∈ F . Suppose that μ(μ + 1) · · · (μ + n) | I. Then NK/Q (Dai ) ≤ NK/Q (I)c . Proof. 2 of [Shl00b]. The only diﬀerences are that we have specialized by taking li = −i, and we have generalized by replacing the element y by an ideal I: this does not aﬀect the proof.
H. Kim and F. W. Roush, Diophantine undecidability of C(t1 , t2 ), J. Algebra 150 (1992), no. 1, 35–44. Mat70. Ju. V. Matijaseviˇc, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282. Maz94. B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), no. 2, 353–371. MB. Laurent Moret-Bailly, paper in preparation, extending results presented in a lecture 18 June 2001 at a conference in honor of Michel Raynaud in Orsay, France.