Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of Pólya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.