We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms β\beta of order nn and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β\beta is a projective s-tensor norm with the sRN property, then for every Asplund space EE, the canonical map ~βn,sE(~βn,sE)\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 Qmin(E)=Q(E)\mathcal{Q}^{min}(E) = \mathcal{Q}(E) for certain polynomial ideal \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 nn-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.Comment: 17 page

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