Polynomial Interpretations over the Reals do not Subsume Polynomial Interpretations over the Integers

Abstract

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.