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

Abstract

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 case 3k² ≢ A (mod p), the gcd has degree one exactly when k²(k²−A)² ≡ Δ (mod p). In the exceptional case 3k² ≡ A (mod p), the gcd has degree two exactly when Δ ≡ 4k⁶ (mod p). We specialize to P(x) = x³ − x² + 1 and show that, for primes outside a finite exceptional set, the number of nonzero shifts producing a nontrivial gcd is either 0 or 6. Using the associated S₃ cubic and Chebotarev's density theorem, we obtain natural density 1/6 for the successful primes. The congruence condition that p mod 23 be a nonzero quadratic residue is necessary but not sufficient; the exact criterion is that 4p = L² + 23M² for some integers L, M.