  We prove a sharp lower bound on the number of terms in an element of the
reduced Gr\"obner basis of a Schubert determinantal ideal $I_w$ under the term
order of [Knutson-Miller '05]. We give three applications. First, we give a
pattern-avoidance characterization of the matrix Schubert varieties whose
defining ideals are binomial. This complements a result of
[Escobar-M\'esz\'aros '16] on matrix Schubert varieties that are toric with
respect to their natural torus action. Second, we give a combinatorial proof
that the recent formulas of [Rajchgot-Robichaux-Weigandt '23] and
[Almousa-Dochtermann-Smith '22] computing the Castelnuovo-Mumford regularity of
vexillary $I_w$ and toric edge ideals of bipartite graphs respectively agree
for binomial $I_w$. Third, we demonstrate that the Gr\"obner basis for $I_w$
given by minimal generators [Gao-Yong '22] is reduced if and only if the
defining permutation $w$ is vexillary.
