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Preprint Number 1897

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1897. Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov
The Diophantine problem in finitely generated commutative rings

Submission date: 17 December 2020


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.
In the remaining case where R has finite rank and zero characteristic, we see that O is a ring of algebraic integers, and then the long-standing conjecture that ℤ is always interpretable by systems of equations in a ring of algebraic integers carries over to R. If true, it implies that the Diophantine problem in R is also undecidable. Thus, in this case the Diophantine problem in every infinite finitely generated commutative unitary ring is undecidable.
The present is the first in a series of papers were we study the Diophantine problem in different types of rings and algebras.

Mathematics Subject Classification: 11D99, 11D72, 11U05

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Full text arXiv 2012.09787: pdf, ps.

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