Reducibility type of polynomials modulo a prime

Let f(x)∈ℤ[x] be a monic polynomial that is irreducible over ℚ , and suppose that (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)∈𝔽_p[x] as f(x)=g_1(x)g_2(x)⋯ g_t(x), where (g_i)=d_i with d_1≤ d_2≤⋯≤ d_t . Let Υ _f:=(U_n)_n≥ 0 be the N th order linear recurrence sequence with initial conditions U_0=U_1=⋯ =U_N-2=0 and U_N-1=1, such that f ( x ) is the characteristic polynomial of Υ _f . In this article, we show, in certain circumstances, how the value modulo p of a particular term of Υ _f determines the reducibility type of f ( x ) modulo p .