For every quadratic field K, deciding solvability of polynomial equations in 16 variables over its ring of integers is undecidable, and 15 variables suffice when K is real quadratic.
On Diophantine equations over $\mathbb Z[i]$ with $52$ unknowns
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In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.
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On Diophantine equations over the integer rings of quadratic fields
For every quadratic field K, deciding solvability of polynomial equations in 16 variables over its ring of integers is undecidable, and 15 variables suffice when K is real quadratic.