In this paper, the authors prove the invariance under simple polynomial maps of a few useful (multivariate) polynomial inequalities. They define the \(q\)-coordinates of polynomials \(p\) and show that, for every \(w\), the values \(R_j(w)\) form the coefficients on the standard monomial basis of the Lagrange interpolation polynomial of \(p\) at the roots of the equation \(q(z)=w\). The purpose of this work is to study this construction, in particular, the definition of \(q\)-coordinates in the higher dimensional case, when \(q\) is a sufficiently simple (so as to enable extending the above decomposition by elementary means) polynomial map from \(\mathbb{C}^n\) to \(\mathbb{C}^n\). The authors apply \(q\)-coordinates to the construction of admissible meshes on the pre-image of a compact set to characterise Bernstein-Markov measures on the pre-image in terms of those on the original set, to show that the pre-image satisfies essentially the same Markov inequalities (as well as other similar inequalities) as its original set. The author give some applications to known inequalities.