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Hilbert’s tenth problem for rings of rational functions. (English) Zbl 1062.03019

Let \(R\) be a non-constant regular semilocal subring of a field of rational functions \(F(t)\) over an algebraically closed field \(F\) of characteristic \(0\). It is proved that the analogue of Hilbert’s tenth problem for \(R\) (does there exist an algorithm which determines the solvability in \(R\) of systems of multivariate polynomial equations with coefficients in \(\mathbb Z[t]\)?) has a negative answer.

MSC:

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