When you express a Calabi Yau manifold as a member of the configuration matrix defined as equation (2.1) of this paper https://arxiv.org/abs/1303.1832 how do you know a priori that every choice of the coefficient of the polynomials, i.e all the members of the configuration, yields a complete intersection manifold (is the intersection necessarily transversal too?).
EDITED: Is the complete intersection condition just equation (2.2)?
For arbitrary choices of the polynomials, the intersection need not be smooth, nor even of the correct dimension.
Their condition (2.2) is one way of saying that the intersection is smooth of the correct dimension. It does depend on the coefficients of the polynomials: the differential $\rm dp_i$ certainly depends on the coefficients of $p_i$.
On the other hand, everything they do in the paper is just working with the multidegrees, never with actual polynomials themselves. They are using the fact that for general polynomials of given multidegrees, the intersection will indeed be smooth. When they say "We demand that...", they mean that they are restricting attention to those collections of polynomials which indeed give a smooth intersection.