Extension of the polynomial Ramsey theorem

Determine whether Corollary 5.10 extends from monomials in x to arbitrary univariate polynomials in x and to arbitrary multivariate polynomials.

Background

Corollary 5.10 classifies Ramsey partition-regular equations of the form axⁿ + P(y) = Q(z), showing that only equations corresponding to multiples of x + y = z or −x + y = z can occur. Its proof relies on expressing the valuation of a monomial in terms of the valuation of its argument, a technique that does not directly apply to general polynomials. The authors explicitly ask whether the result can be extended first to arbitrary univariate polynomials and then to multivariate polynomials.