Past exam of the geometry course of the University of Oxford / 2010 / B / 5 / b / Solution

For curves with no common local component at $p$,In local affine coordinates, their defining equations have linearly independent gradients exactly in this case. These independent linear terms form local coordinates, so the local intersection quotient has dimension one. Conversely, if either equation lacks a nonzero linear term or the two linear terms are dependent, the local intersection quotient cannot have dimension one. This is transverse intersection of plane curves of two smooth plane curves.