Prime scaling changes the geometric bad-prime factor by at most an absolute constant. Outside the scaling prime, the homogeneous models are related by an invertible change of coordinates and a nonzero scalar.

theorem CCDN.hypersurfaceBadPrimeFactor_scaledHomogenize_le {n : ℕ} (f : MvPolynomial (Fin n) ℤ) (hf : f ≠ 0) {p : ℕ} (hp : Nat.Prime p) : hypersurfaceBadPrimeFactor (scaledHomogenize f.totalDegree (↑p) f) ≤ Real.exp 1 * hypersurfaceBadPrimeFactor (scaledHomogenize f.totalDegree 1 f)

Prime scaling preserves all geometric bad primes away from the scaling prime, and therefore multiplies the paper's bad-prime factor by at most e.