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Hilbert’s Tenth problem for function fields of varieties over number fields and \(p\)-adic fields. (English) Zbl 1152.11050
Hilbert’s tenth problem (HTP) for a commutative ring \(R\) with coefficients in a recursive subring \(A\) of \(R\) is the question of the existence of an algorithm which, given any polynomial equation with coefficients in \(A\), determines whether the equation has solutions in \(R\).
Let \(k\) be a subfield of a finite field extension of \(\mathbb{Q}_p\) with \(p\) odd, and let \(R\) be a finite field extension of a rational function field \(k(t, t_2, \dots, t_n)\). The author of the paper under review proves that there exist finitely many elements \(c_1, \dots, c_m\) of \(k(t)\), not all constants, such that HTP for \(R\) with coefficients in \(A := \mathbb{Z}[c_1, \dots, c_m]\) is undecidable.
To obtain this very interesting result as a consequence of the undecidability of the existential theory of the structure \((\mathbb{Z}, 0, 1, +, \cdot)\), the author constructs a diophantine model of \((\mathbb{Z}, 0, 1, +, \cdot)\) over \(R\) with coefficients in \(A\) using and extending some basic results due to J. Denef [Trans. Am. Math. Soc. 242, 391–399 (1978; Zbl 0399.10048)], K. H. Kim, F. W. Roush [J. Algebra 176, No. 1, 83–110 (1995; Zbl 0858.12006)], and L. Moret-Bailly [J. Reine Angew. Math. 587, 77–143 (2005; Zbl 1085.14029)] concerning elliptic curves and HTP for algebraic function fields.

MSC:
11U05 Decidability (number-theoretic aspects)
03C07 Basic properties of first-order languages and structures
14G05 Rational points
11R58 Arithmetic theory of algebraic function fields
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