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Diophantine undecidability of function fields of characteristic greater than 2, finitely generated over fields algebraic over a finite field. (English) Zbl 1011.03026

Let \(C\) be field, algebraic over a finite field of characteristic \(p>2\). Assume that \(C\) has an extension of degree \(p\). Let \(F\) be a function field of one variable over a finitely generated extension of \(C\). Then the analogue of Hilbert’s tenth problem has a negative solution over \(F\), meaning that there is a finite set of elements of \(F\) which generate a ring \(B\) with the following property: there is no algorithm which detects the existence of solutions in \(F\) of systems of polynomial equations (in any number of variables) with coefficients in \(B\). The set \(B\) apparently contains at least one transcendental element and depends on \(F\). This result increases the class of fields for which a similar statement has been proved. The expectation is that a similar result may hold for any finitely generated function field.

MSC:

03C60 Model-theoretic algebra
11U05 Decidability (number-theoretic aspects)
03D35 Undecidability and degrees of sets of sentences
03B25 Decidability of theories and sets of sentences
12L05 Decidability and field theory
14G25 Global ground fields in algebraic geometry
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