Nair–Tenenbaum bounds uniform with respect to the discriminant

For functions F satisfying a certain submultiplicativity condition and polynomials Q 1 , . . ., Q k in [ X ], Nair and Tenenbaum obtained an upper bound on the short sum ∑x < n ≤ x+y F(|Q₁(n)|,,|Qₖ(n)|) with an implicit dependency on the discriminant of Q 1 . . . Q k . We obtain a similar upper bound uniform in the discriminant.