The polynomial dual of an operator ideal

We prove that the adjoint of a continuous homogeneous polynomial $P$ between Banach spaces belongs to a given operator ideal $\mathcal I$ if and only if $P$ admits a factorization $P = u \circ Q$ where the adjoint of the linear operator $u$ belongs to $\mathcal I$ . Several consequences of this factorization are obtained, for example we characterize the polynomials whose adjoints are absolutely $p$ -summing.