The symmetric Radon-Nikod\'ym property for tensor norms

We introduce the symmetric-Radon-Nikod'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}{\beta}^{n,s} E’ \to \Big(\widetilde{\otimes}{\beta’}^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $$\mathcal{Q}$^{min}(E) = $\mathcal{Q}$(E)$forcertainpolynomialideal$\Q$. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of $n$-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund spaces.