A correspondence between Hilbert polynomials and Chern polynomials over projective spaces

We construct a map ζ from K₀(ℙ^d) to (ℤ[x]/x^d+1)^x × ℤ, where (ℤ[x]/x ^d+1)^x is a multiplicative Abelian group with identity 1, and show that ζ; induces an isomorphism between K₀ (ℙ ^d) and its image. This is inspired by a correspondence between Chern and Hilbert polynomials stated in Eisenbud [1, Exercise 19.18]. The equivalence of these two polynomials over ℙ^d is discussed in this paper.