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On diophantine definability and decidability in some rings of algebraic functions of characteristic 0. (English) Zbl 1011.03027

Let \(K\) be a function field of one variable over a field of constants which has finite transcendence degree over the field of complex numbers. Let \(M\) be a finite extension of \(K\) and let \(W\) be a set of primes of \(K\) such that only a finite number of elements of \(W\) split in \(M\). Then there is a set \(W'\) of primes of \(K\) so that “the analogue of Hilbert’s tenth problem has a negative solution” over the ring \({\mathcal O}_{K,W'}\), of elements of \(K\) integral modulo all primes of \(K\), except possibly for elements of \(W'\). The quoted phrase means the following: There is a finite set of elements of \(K\) which generate a ring \(B\) with the property that there is no algorithm which detects the existence of solutions in \(O_{K,W'}\) of systems of polynomial equations (in any number of variables) with coefficients in \(B\). The set \(B\) depends on \(K\), \(M\) and \(W\). The author applies the above to the case that the field of constants is finitely generated over the rational numbers and the elements of the set \(W\) are of bounded degree. In this case the result is stronger: the rational integers are existentially definable.

MSC:

03C60 Model-theoretic algebra
11U05 Decidability (number-theoretic aspects)
12L05 Decidability and field theory
03B25 Decidability of theories and sets of sentences
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