Reducibility type of polynomials modulo a prime

Let $$f(x)\in {\mathbb Z}[x]$$ f ( x ) ∈ Z [ x ] be a monic polynomial that is irreducible over $${\mathbb Q}$$ Q , and suppose that $$\deg (f)=N\ge 2$$ 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 $$(d_1,d_2,\ldots ,d_t)_p$$ ( d 1 , d 2 , … , d t ) p if f(x) factors into distinct irreducibles $$g_i(x)\in {\mathbb F}_p[x]$$ g i ( x ) ∈ F p [ x ] as $$\begin{aligned} f(x)=g_1(x)g_2(x)\cdots g_t(x), \end{aligned}$$ f ( x ) = g 1 ( x ) g 2 ( x ) ⋯ g t ( x ) , where $$\deg (g_i)=d_i$$ deg ( g i ) = d i with $$d_1\le d_2\le \cdots \le d_t$$ d 1 ≤ d 2 ≤ ⋯ ≤ d t . Let $$\Upsilon _f:=(U_n)_{n\ge 0}$$ Υ f : = ( U n ) n ≥ 0 be the Nth order linear recurrence sequence with initial conditions $$\begin{aligned} U_0=U_1=\cdots =U_{N-2}=0 \quad \text{ and } \quad U_{N-1}=1, \end{aligned}$$ U 0 = U 1 = ⋯ = U N - 2 = 0 and U N - 1 = 1 , such that f(x) is the characteristic polynomial of $$\Upsilon _f$$ Υ f . In this article, we show, in certain circumstances, how the value modulo p of a particular term of $$\Upsilon _f$$ Υ f determines the reducibility type of f(x) modulo p.