Abstract We study P w ∗ ( n X∗, Y ), the Banach space of all n −homogeneous polynomials which are w∗ − w −continuous, endowed with the sup norm. The following facts are proved: (1) c 0 (Γ) embeds in Pw ∗ ( n X∗, Y ) iff either ℓ ∞ (Γ) embeds in Pw ∗ ( n X∗, Y ) or c 0 (Γ) embeds in the sum X ⊕ Y . (2) If c 0 (Γ) is isomorphic to a complemented subspace of Pw ∗ ( n X∗, Y ), then c 0 (Γ) embeds in X ⊕ Y . We also extend some linear techniques in order to embed complementably c 0 (Γ) in either Pw ∗ ( n X∗, Y ) or P( n X , Y ), the Banach space of all n −homogeneous polynomials from X to Y , endowed with the sup norm. Finally, if Pw ∗K( n X ∗, Y ) is the closed subspace of all compact polynomials in Pw∗ ( n X ∗, Y ), we prove that ℓ ∞ embeds in Pw∗K ( n X ∗, Y ) iff ℓ ∞ embeds in either X or Y . As a consequence, we prove that if c0 embeds in Pw ∗ K ( n X ∗, Y ), then Pw ∗ K ( n X ∗, Y ) = n X ∗ ( n X ∗, Y ) iff only one of the following statements is true: (1) c 0 embeds in Y and X has the Schur property, (2) c 0 embeds in X and Y has the Schur property. 2020 Mathematics Subject Classification. 46B03, 46B25, 46E40, 46G25.