When does a Noetherian commutative ring R R have uniform symbolic topologies on primes–read, when does there exist an integer D > 0 D>0 such that the symbolic power P ( D r ) ⊆ P r P^{(Dr)} \subseteq P^r for all prime ideals P ⊆ R P \subseteq R and all r > 0 r >0 ? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then search for analogues where the ring R R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal Q Q inside of tensor products of domains of finite type over an algebraically closed field F \mathbb {F} , we present binomial and multinomial expansion criteria for containments of type Q ( E r ) ⊆ Q r Q^{(E r)} \subseteq Q^r , or even better, of type Q ( E ( r − 1 ) + 1 ) ⊆ Q r Q^{(E (r-1)+1)} \subseteq Q^r for all r > 0 r>0 . The final section consolidates remarks on how often we can utilize these criteria, presenting an example.