Nair–Tenenbaum bounds uniform with respect to the discriminant

For functionsFsatisfying a certain submultiplicativity condition and polynomialsQ1, . . .,Qkin[X], Nair and Tenenbaum obtained an upper bound on the short sum$\sum_{x < n \leq x+y} F(|Q_{1}(n)|,\dotsc,|Q_{k}(n)|)$with an implicit dependency on the discriminant ofQ1. . .Qk. We obtain a similar upper bound uniform in the discriminant.