Effective hyperplane avoidance from the universal Noether forms. The polynomial is nonzero by a constructed geometrically good section. It is homogeneous of degree d(d²−1)+1 and vanishes on every bad graph section, as well as on the omitted chart.

theorem CCDN.exists_effective_hyperplane_avoidance {n : ℕ} (f : MvPolynomial (Option (Option (Fin (2 + n)))) ℤ) (d : ℕ) (hd : 2 ≤ d) (hf : f.IsHomogeneous d) (hir : AbsolutelyIrreducible f) : ∃ (G : MvPolynomial (Option (Option (Fin (2 + n)))) ℤ), G ≠ 0 ∧ G.IsHomogeneous (d * (d ^ 2 - 1) + 1) ∧ G.totalDegree ≤ (4 + n) * d * (d ^ 2 - 1) ∧ ∀ (K : Type u) [inst : Field K] (A : Option (Option (Fin (2 + n))) → K), (A none ≠ 0 → ¬Irreducible ((MvPolynomial.map (algebraMap K (AlgebraicClosure K))) ((graphHyperplaneRestriction fun (i : Option (Fin (2 + n))) => -A (some i) / A none) ((MvPolynomial.map (Int.castRingHom K)) f)))) → MvPolynomial.eval₂ (Int.castRingHom K) A G = 0

A corrected form of CCDN Lemma 4.3.7 for degree d ≥ 2, in explicit hyperplane coordinates: for an absolutely irreducible integer form f of degree d, there is a nonzero integer form G in the hyperplane coefficients, homogeneous of degree d * (d ^ 2 - 1) + 1 and of total degree at most (4 + n) * d * (d ^ 2 - 1), that vanishes at every coefficient vector A over any field, either with A₀ = 0 or for which the restriction of f to the hyperplane A is not absolutely irreducible.