Sum the sharp good-section bound and the separately controlled bad and zero contributions over the full plane parameter space.
theorem
CCDN.plane_sections_incidence_bound
{σ : Type u_1}
[Fintype σ]
(p : ℕ)
[Fact (Nat.Prime p)]
(f : MvPolynomial σ (ZMod p))
(hsize : stepanovThreshold ↑f.totalDegree < ↑p)
:
↑(Nat.card { z : σ → ZMod p // (MvPolynomial.eval z) f = 0 }) * ↑p ^ (2 * Fintype.card σ + 2) ≤ ↑p ^ (3 * Fintype.card σ) * (↑p + planeResidueConstant * ↑f.totalDegree ^ 2 * √↑p) + ↑(Nat.card (BadPlaneParameters f)) * (↑f.totalDegree * ↑p) + ↑(Nat.card (ZeroPlaneParameters f)) * ↑p ^ 2
The exact incidence identity with each section bounded according to its equation.