Uppers to zero in intersections of prime ideals

Let R be a Noetherian integral domain. The structure of the partially-ordered set of prime ideals of R[z], the polynomial ring in one indeterminate over R, is not fully understood. I demonstrate that if p₁,…,p_n are prime ideals in R[x] with ht(p_i) > 2 and either n = 1 or R is not a Henselian local domain of dimension < 2, then pi D-o-Cpn contains [R] many prime ideals which intersect R at (0). I also show that if R is a Noetherian domain that is not a Henselian local domain and p₁,…,p_n are prime ideals with height > 2 each of which contains a monic polynomial, then their intersection contains [R] many prime ideals meeting R at (0), each containing a monic polynomial.