Polynomials with an integral representation

We introduce a large class of polynomials between Banach spaces which admit an integral representation. This class coincides with the class of integral polynomials in the scalar-valued case, but it is larger in the vector-valued case. By comparing this class with other well-known classes, we obtain characterizations of Banach spaces containing no copy of c0, of Grothendieck spaces, and of L∞-spaces. Among some other results, we show that a polynomial has an integral representation if and only if it admits an orthogonally additive extension to C(BE⁎). Moreover, if the range space is complemented in its bidual, every polynomial with an integral representation is extendible to every superspace.