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Undecidability of existential theories of rings and fields: A survey. (English) Zbl 0981.03013

Denef, Jan (ed.) et al., Hilbert’s tenth problem: relations with arithmetic and algebraic geometry. Proceedings of the workshop, Ghent University, Belgium, November 2-5, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 270, 49-105 (2000).
The aim of the reviewed paper is to give an overview of results and problems connected with Hilbert’s tenth problem for various rings and fields.
Let \(R\) be a ring, \(L\) be a first-order language of ring theory which includes symbols for some elements of \(R\). A diophantine polynomial in \(L\) over \(R\) is a polynomial whose coefficients are in the subring of \(R\) which is generated by such elements. If \(P\) is a diophantine polynomial in \(L\), \(P=0\) is a diophantine equation in \(L\).
The emphasis is on the decidability problem for existential and diophantine theories of rings and fields of algebraic and meromorphic functions.
For the entire collection see [Zbl 0955.00034].

MSC:

03B25 Decidability of theories and sets of sentences
11U05 Decidability (number-theoretic aspects)
12L05 Decidability and field theory
30E99 Miscellaneous topics of analysis in the complex plane
32A99 Holomorphic functions of several complex variables
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