We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our main theorem gives two simple conditions under which the characteristic polynomial factors in this way. We will see that Stanley’s Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate the factorization of a poly...

Let f(x) = ∑d s=0 asx s ∈ Z[x] be a polynomial with ad 6≡ 0 mod p. Take z ∈ Fp and let Oz = {fi(z)}i∈Z+ ⊂ Fp be the orbit of z under f , where fi(z) = f(fi−1(z)) and f0(z) = z. For M < |Oz|, we study the diameter of the partial orbit Oz,M = {z, f(z), f2(z), . . . , fM−1(z)} and prove that diam Oz,M & min { M c log logM , Mp , M 1 2p 1 2 } , where ‘diameter’ is naturally defined in Fp and c depe...