Reducibility type of polynomials modulo a prime

Let f (x) is an element of Z[x] be a monic polynomial that is irreducible over Q, and suppose that deg(f) = N >= 2. For a prime p not dividing the discriminant of f (x), we define the reducibility type of f (x) modulo p to be (d(1), d(2), . . . , d(t)) (p) if f (x) factors into distinct irreducibles g(i) (x) is an element of F-p[x] as f (x) = g(1)(x)g(2)(x) center dot center dot center dot g(t)(x), where deg(g(i)) = d(i) with d(1) <= d(2) <= center dot center dot center dot <= d(t). Let Upsilon(f) := (U-n)(n >= 0) be the Nth order linear recurrence sequence with initial conditions U-0 = U-1 = center dot center dot center dot = UN-2 = 0 and UN-1 = 1, such that f (x) is the characteristic polynomial of Upsilon(f). In this article, we show, in certain circumstances, how the value modulo p of a particular term of Upsilon(f) determines the reducibility type of f (x) modulo p.