Aditya Kumar

Integer Translates of a Monic Cubic: Resultants and Common Factors over Finite Fields

Let f(x) = x³ + ax² + bx + c ∈ Z[x] be a monic cubic, put A = a² − 3b, Δ = disc(f). We derive the translate-resultant identity Res_x(f(x), f(x+k)) = k³(k²(k²−A)² − Δ). We then classify the degree of gcd in F_p[x] of (f(x), f(x+k)) for primes p ≠ 2,3 with p ∤ k. In the generic cas...