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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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References:
 [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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