Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.