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This is also not really an answer, but it explains an alternative version of the characterisation that I find easier to work with. Let $P_d(R)$ be the polynomial ring over $R$ in variables $x_0,\dotsc,x_d$, and order the monomials lexicographically. Say that $f\in P_d(R)$ is comonic if the lowest monomial has coefficient one. Say that an $R$-algebra homomorphism $\phi\colon P_d(R)\to R$ is thin if the kernel contains a comonic polynomial. After a little translation, the results of Coquand and Lombardi say that $R$ has dimension $\leq d$ iff every such homomorphism is thin.