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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $ma{x}_{\omega \in }\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$. We study the stability of the APDP by c₀-, ${\ell }_{\infty }$- and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty }(\mu ,X)$ and C(K,X), where X has the APDP.