Specializations of indecomposable polynomials

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial P (x) ∈ Z[x] to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t1, . . . , tr, x) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p = 0 or p > deg(f), then the one variable specialized polynomial f(t1 + α ∗ 1x, . . . , t ∗ r + α ∗ rx, x) is indecomposable for all (t1, . . . , t ∗ r , α ∗ 1, . . . , α ∗ r) ∈ k 2r outside a proper Zariski closed subset.