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Preprint Number 1897
1897. Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov The Diophantine problem in finitely generated commutative rings E-mail: Submission date: 17 December 2020 Abstract: We study systems of polynomial equations in infinite finitely generated
commutative associative rings with an identity element. For each such
ring R
we obtain an interpretation by systems of equations of a ring of
integers O
of a finite field extension of either ℚ or 𝔽_p(t), for
some prime p and variable t. This implies that the Diophantine problem
(decidability of systems of polynomial equations) in O is reducible to the
same problem in R. If, in particular, R has positive characteristic or,
more generally, if R has infinite rank, then we further obtain an
interpretation by systems of equations of the ring 𝔽_p[t] in R.
This implies that the Diophantine problem in R is undecidable in this
case. Mathematics Subject Classification: 11D99, 11D72, 11U05 Keywords and phrases: |

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