Polynomial interpretations are a useful technique for proving termination

of term rewrite systems. They come in various flavors:

polynomial interpretations with real, rational and integer coefficients.

In 2006, Lucas proved that there are rewrite systems that can be shown

polynomially terminating by polynomial interpretations with

real (algebraic)

coefficients, but cannot be shown polynomially terminating using

polynomials with rational coefficients only.

He also proved a similar theorem with respect to the use of

rational coefficients versus integer coefficients.

In this paper we show that polynomial interpretations with real or

rational coefficients do not subsume polynomial interpretations with

integer coefficients, contrary to what is commonly believed.

We further show that polynomial interpretations with real

coefficients subsume polynomial interpretations with rational

coefficients.