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### Abstract

We formalize several undecidability results on termination
for

*one-rule*term rewrite systems by means of simple reductions from Hilbert's 10th problem. To be more precise, for a class \(C\) of reduction orders, we consider the question for a given rewrite rule \(\ell \to r\), whether there is some reduction order \({\succ} \in C\) such that \(\ell \succ r\). We include undecidability results for each of the following classes \(C\):- the class of
*linear*polynomial interpretations over the natural numbers, - the class of linear polynomial interpretations over
the natural numbers in the
*weakly monotone*setting, - the class of Knuth–Bendix orders with
*subterm coefficients*, - the class of
*non-linear*polynomial interpretations over the natural numbers, and - the class of non-linear polynomial interpretations over the
*rational*and*real*numbers.