Linear Termination over N is Undecidable

Recently it was shown that it is undecidable whether a term rewrite system can be proved terminating by a polynomial interpretation in the natural numbers. In this paper we show that this is also the case when restricting the interpretations to linear polynomials, as is often done in tools, and when only considering single-rule rewrite systems. What is more, the new undecidability proof is simpler than the previous one. We further show that polynomial termination over the rationals/reals is undecidable.