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

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.