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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.

11U05 Decidability (number-theoretic aspects)
03C07 Basic properties of first-order languages and structures
14G05 Rational points
11R58 Arithmetic theory of algebraic function fields
Full Text: DOI
[1] (), with the support of the International Mathematical Union
[2] Conrad, Brian, Gross – zagier revisited, (), 67-163, with an appendix by W.R. Mann · Zbl 1072.11040
[3] Cremona, J.E., Algorithms for modular elliptic curves, (1997), Cambridge Univ. Press Cambridge · Zbl 0872.14041
[4] Davis, Martin; Putnam, Hilary; Robinson, Julia, The decision problem for exponential Diophantine equations, Ann. of math. (2), 74, 425-436, (1961) · Zbl 0111.01003
[5] Denef, Jan, The Diophantine problem for polynomial rings and fields of rational functions, Trans. amer. math. soc., 242, 391-399, (1978) · Zbl 0399.10048
[6] Deuring, Max, Lectures on the theory of algebraic functions of one variable, Lecture notes in math., vol. 314, (1973), Springer-Verlag Berlin · Zbl 0249.14008
[7] Eisenträger, Kirsten, Hilbert’s tenth problem for algebraic function fields of characteristic 2, Pacific J. math., 210, 2, 261-281, (2003) · Zbl 1057.11067
[8] Eisenträger, Kirsten, Hilbert’s tenth problem for function fields of varieties over \(\mathbb{C}\), Int. math. res. not., 59, 3191-3205, (2004) · Zbl 1109.11061
[9] Kim, K.H.; Roush, F.W., Diophantine undecidability of \(\mathbf{C}(t_1, t_2)\), J. algebra, 150, 1, 35-44, (1992) · Zbl 0754.11039
[10] Kim, K.H.; Roush, F.W., Diophantine unsolvability over p-adic function fields, J. algebra, 176, 1, 83-110, (1995) · Zbl 0858.12006
[11] Lam, T.Y., The algebraic theory of quadratic forms, (1980), Benjamin/Cummings Publishing Co. Inc., Advanced Book Program Reading, MA, revised second printing, Mathematics Lecture Note Series · Zbl 0437.10006
[12] Matijasevič, Yu.V., The diophantineness of enumerable sets, Dokl. akad. nauk SSSR, 191, 279-282, (1970)
[13] Moret-Bailly, Laurent, Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and p-adic fields, J. reine angew. math., 587, 77-143, (2005) · Zbl 1085.14029
[14] Pheidas, Thanases, Hilbert’s tenth problem for fields of rational functions over finite fields, Invent. math., 103, 1, 1-8, (1991) · Zbl 0696.12022
[15] Pheidas, Thanases, Extensions of Hilbert’s tenth problem, J. symbolic logic, 59, 2, 372-397, (1994) · Zbl 0812.11071
[16] Rosen, Michael, Number theory in function fields, Grad. texts in math., vol. 210, (2002), Springer-Verlag New York · Zbl 1043.11079
[17] Shlapentokh, Alexandra, Hilbert’s tenth problem for algebraic function fields over infinite fields of constants of positive characteristic, Pacific J. math., 193, 2, 463-500, (2000) · Zbl 1010.12008
[18] Silverman, Joseph, Heights and the specialization map for families of abelian varieties, J. reine angew. math., 342, 197-211, (1983) · Zbl 0505.14035
[19] Silverman, Joseph H., The arithmetic of elliptic curves, Grad. texts in math., vol. 106, (1994), Springer-Verlag New York, corrected reprint of the 1986 original · Zbl 0911.14015
[20] Videla, Carlos R., Hilbert’s tenth problem for rational function fields in characteristic 2, Proc. amer. math. soc., 120, 1, 249-253, (1994) · Zbl 0795.03015
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