In this paper, we study the ideal of strongly $(p,\sigma )$-continuous polynomials that extends the nowadays well-known ideal of Cohen strongly $p$-summing polynomials to the more general setting of the interpolated ideals of polynomials. We give an analogue of the Pietsch domination theorem among other results. Also, we introduce the polynomial ideal of factorable strongly $(p,\sigma,q,\nu )$-nuclear polynomials in order to characterize those polynomials whose linearizations are $(p,\sigma,q,\nu )$-nuclear.
