@[reducible, inline]

abbrev Lec5BezoutPascal.P2 (k : Type u_1) [Field k] : Type u_1

The projective plane P^2 over k, identified with the projectivization of k^3.

def Lec5BezoutPascal.P2.liesOn (k : Type u_1) [Field k] (p : P2 k) (f : MvPolynomial (Fin 3) k) : Prop

A point p ∈ P^2 lies on the curve defined by a polynomial f if f vanishes on a (any) representative of p.

noncomputable def Lec5BezoutPascal.translatePoly (k : Type u_1) [Field k] {n : ℕ} (g : MvPolynomial (Fin n) k) (p : Fin n → k) : MvPolynomial (Fin n) k

Translate of a polynomial g by a point p: the polynomial obtained by substituting x_i ↦ x_i + p_i.

theorem Lec5BezoutPascal.translatePoly_injective (k : Type u_1) [Field k] {n : ℕ} (p : Fin n → k) : Function.Injective fun (g : MvPolynomial (Fin n) k) => translatePoly k g p

Translation of polynomials by a fixed point is an injective operation, since translating by -p is a left inverse.

theorem Lec5BezoutPascal.translatePoly_ne_zero (k : Type u_1) [Field k] {n : ℕ} (g : MvPolynomial (Fin n) k) (p : Fin n → k) (hg : g ≠ 0) : translatePoly k g p ≠ 0

Translating a nonzero polynomial yields a nonzero polynomial.

theorem Lec5BezoutPascal.exists_homogeneousComponent_ne_zero (k : Type u_1) [Field k] {n : ℕ} (φ : MvPolynomial (Fin n) k) (h : φ ≠ 0) : ∃ (m : ℕ), (MvPolynomial.homogeneousComponent m) φ ≠ 0

A nonzero polynomial has a nonzero homogeneous component of some degree.

noncomputable def Lec5BezoutPascal.hypersurfaceMultiplicity (k : Type u_1) [Field k] {n : ℕ} (g : MvPolynomial (Fin n) k) (p : Fin n → k) (hg : g ≠ 0) : ℕ

Multiplicity of a hypersurface V(g) at a point p: the smallest degree at which the translate g(x + p) has a nonzero homogeneous component.

theorem Lec5BezoutPascal.hypersurfaceMultiplicity_spec (k : Type u_1) [Field k] {n : ℕ} (g : MvPolynomial (Fin n) k) (p : Fin n → k) (hg : g ≠ 0) : (MvPolynomial.homogeneousComponent (hypersurfaceMultiplicity k g p hg)) (translatePoly k g p) ≠ 0

At the multiplicity, the corresponding homogeneous component of g(x + p) is nonzero.

theorem Lec5BezoutPascal.hypersurfaceMultiplicity_min (k : Type u_1) [Field k] {n : ℕ} (g : MvPolynomial (Fin n) k) (p : Fin n → k) (hg : g ≠ 0) (m : ℕ) (hm : m < hypersurfaceMultiplicity k g p hg) : (MvPolynomial.homogeneousComponent m) (translatePoly k g p) = 0

Below the multiplicity, every homogeneous component of the translated polynomial vanishes.

@[reducible, inline]

abbrev Lec5BezoutPascal.CurveQuotient (k : Type u_1) [Field k] (f g : MvPolynomial (Fin 2) k) : Type u_1

The coordinate ring of the intersection scheme of two affine plane curves V(f) and V(g): the quotient k[x, y] / (f, g).

noncomputable def Lec5BezoutPascal.mulByVarOperator (k : Type u_1) [Field k] (f g : MvPolynomial (Fin 2) k) (i : Fin 2) : Module.End k (CurveQuotient k f g)

Multiplication-by-x_i operator on the curve quotient k[x, y]/(f, g), viewed as a k-linear endomorphism.

noncomputable def Lec5BezoutPascal.intersectionMultiplicity (k : Type u_1) [Field k] (f g : MvPolynomial (Fin 2) k) (p : Fin 2 → k) [Module.Finite k (CurveQuotient k f g)] : ℕ

Intersection multiplicity of two affine plane curves at a point p: the dimension of the joint generalized eigenspace of the multiplication-by-coordinate operators at the eigenvalue p.

structure Lec5BezoutPascal.PlaneCurve (k : Type u_1) [Field k] : Type u_1

A plane projective curve in P^2: a nonzero homogeneous polynomial in three variables of a given degree.

• poly : MvPolynomial (Fin 3) k
• poly_ne_zero : self.poly ≠ 0
• deg : ℕ
• poly_homogeneous : self.poly.IsHomogeneous self.deg

def Lec5BezoutPascal.PlaneCurve.vanishingLocus (k : Type u_1) [Field k] (C : PlaneCurve k) : Set (P2 k)

The vanishing locus of a plane curve C as a subset of P^2.

def Lec5BezoutPascal.PlaneCurve.noCommonComponent (k : Type u_1) [Field k] (C₁ C₂ : PlaneCurve k) : Prop

Two plane curves have no common irreducible component iff their defining polynomials are coprime.

def Lec5BezoutPascal.PlaneCurve.intersectionPoints (k : Type u_1) [Field k] (C₁ C₂ : PlaneCurve k) : Set (P2 k)

The set of intersection points of two plane curves: the intersection of their vanishing loci in P^2.

theorem Lec5BezoutPascal.bezout_intersection_finite (k : Type u_2) [Field k] [IsAlgClosed k] (C₁ C₂ : PlaneCurve k) (hcoprime : PlaneCurve.noCommonComponent k C₁ C₂) : ∃ (S : Finset (P2 k)), ↑S = PlaneCurve.intersectionPoints k C₁ C₂

Finiteness part of Bezout: two plane curves with no common component meet in a finite set of points in P^2.

theorem Lec5BezoutPascal.bezout_multiplicity_sum (k : Type u_2) [Field k] [IsAlgClosed k] (C₁ C₂ : PlaneCurve k) (hcoprime : PlaneCurve.noCommonComponent k C₁ C₂) (S : Finset (P2 k)) (hS : ↑S = PlaneCurve.intersectionPoints k C₁ C₂) : ∃ (mult : P2 k → ℕ), (∀ p ∈ S, 0 < mult p) ∧ (∀ p ∉ S, mult p = 0) ∧ S.sum mult = C₁.deg * C₂.deg

Multiplicity-sum part of Bezout: counted with multiplicities, the intersection points of two plane curves without common component sum to the product of their degrees.

theorem Lec5BezoutPascal.bezout_theorem_5_2 (k : Type u_2) [Field k] [IsAlgClosed k] (C₁ C₂ : PlaneCurve k) (hcoprime : PlaneCurve.noCommonComponent k C₁ C₂) : ∃ (S : Finset (P2 k)) (mult : P2 k → ℕ), ↑S = PlaneCurve.intersectionPoints k C₁ C₂ ∧ (∀ p ∈ S, 0 < mult p) ∧ (∀ p ∉ S, mult p = 0) ∧ S.sum mult = C₁.deg * C₂.deg

Lecture 5, Theorem 5.2 (Bezout): for two plane projective curves over an algebraically closed field without common component, the intersection is a finite set whose multiplicities sum to the product of the degrees.

theorem Lec5BezoutPascal.bezout_projective (k : Type u_2) [Field k] (d e : ℕ) (hd : 0 < d) (he : 0 < e) (f g : MvPolynomial (Fin 3) k) (hf_homog : f.IsHomogeneous d) (hg_homog : g.IsHomogeneous e) (hf_ne : f ≠ 0) (hg_ne : g ≠ 0) (hcoprime : IsCoprime f g) (S : Finset (ProjectivePoint k f g)) (hS : ∀ (p : ProjectivePoint k f g), projLocalIntersectionMultiplicity k f g p ≠ 0 → p ∈ S) (hS' : ∀ p ∈ S, projLocalIntersectionMultiplicity k f g p ≠ 0) : ∑ p ∈ S, projLocalIntersectionMultiplicity k f g p = d * e

Projective form of Bezout (multiplicity-sum statement): the sum of local intersection multiplicities of two coprime homogeneous polynomials in three variables equals the product of their degrees.

def Lec5BezoutPascal.P2Collinear (k : Type u_1) [Field k] (p q r : P2 k) : Prop

Three points in P^2 are collinear if some nonzero linear form vanishes at all of them.

def Lec5BezoutPascal.IsConic (k : Type u_1) [Field k] (C : PlaneCurve k) : Prop

A plane curve is a conic if its degree is 2.

noncomputable def Lec5BezoutPascal.lineThroughPoints (k : Type u_1) [Field k] (p q : P2 k) : MvPolynomial (Fin 3) k

Defining linear form of the line through two points of P^2, given by the cross product of their representatives.

structure Lec5BezoutPascal.PascalInscribedHexagon (k : Type u_1) [Field k] : Type u_1

Data of a hexagon inscribed in a conic: the conic, six distinct vertices on it, indexed cyclically by Fin 6.

• conic : PlaneCurve k
• is_conic : IsConic k self.conic
• vertex : Fin 6 → P2 k
• vertices_on_conic (i : Fin 6) : P2.liesOn k (self.vertex i) self.conic.poly
• vertices_distinct : Function.Injective self.vertex

def Lec5BezoutPascal.hexNext (i : Fin 6) : Fin 6

Cyclic successor on Fin 6, used to index the sides of an inscribed hexagon.

def Lec5BezoutPascal.hexOpposite (i : Fin 6) : Fin 6

Opposite vertex map on Fin 6 (shift by 3), used to pair opposite sides of an inscribed hexagon.

noncomputable def Lec5BezoutPascal.PascalInscribedHexagon.side (k : Type u_1) [Field k] (H : PascalInscribedHexagon k) (i : Fin 6) : MvPolynomial (Fin 3) k

The i-th side of an inscribed hexagon: the line through the i-th and (i + 1)-th vertices.

theorem Lec5BezoutPascal.bezout_excess_intersection_implies_divisibility (k : Type u_2) [Field k] [IsAlgClosed k] (q f : MvPolynomial (Fin 3) k) (_hq_irred : Irreducible q) (_hq_homog : q.IsHomogeneous 2) (_hf_ne : f ≠ 0) (_hf_homog : f.IsHomogeneous 3) (S : Finset (P2 k)) (_hS_card : 6 < S.card) (_hS_on_q : ∀ p ∈ S, P2.liesOn k p q) (_hS_on_f : ∀ p ∈ S, P2.liesOn k p f) : q ∣ f

Consequence of Bezout: if an irreducible conic q and a cubic f share more than 6 points, then q divides f.

theorem Lec5BezoutPascal.isHomogeneous_of_homogeneous_dvd (k : Type u_2) [Field k] (σ : Type u_3) [DecidableEq σ] (q L : MvPolynomial σ k) (m n : ℕ) : q ≠ 0 → q.IsHomogeneous m → (q * L).IsHomogeneous n → m ≤ n → L.IsHomogeneous (n - m)

If q is homogeneous of degree m and q * L is homogeneous of degree n ≥ m, then L is itself homogeneous of degree n - m.

theorem Lec5BezoutPascal.conic_divides_cubic_gives_line (k : Type u_2) [Field k] (q f : MvPolynomial (Fin 3) k) (_hq_ne : q ≠ 0) (_hq_homog : q.IsHomogeneous 2) (_hf_homog : f.IsHomogeneous 3) (_hdvd : q ∣ f) : ∃ (L : MvPolynomial (Fin 3) k), L.IsHomogeneous 1 ∧ f = q * L ∧ ∀ (p : P2 k), P2.liesOn k p f ↔ P2.liesOn k p q ∨ P2.liesOn k p L

If a conic divides a cubic, then the cubic factors as the conic times a linear form, and the cubic vanishes at a point iff the conic or the line does.

theorem Lec5BezoutPascal.pascal_cubic_construction (k : Type u_2) [Field k] [IsAlgClosed k] (H : PascalInscribedHexagon k) (_hconic_irred : Irreducible H.conic.poly) : ∃ (P₁ : P2 k) (P₂ : P2 k) (P₃ : P2 k) (P_cubic : MvPolynomial (Fin 3) k) (p₇ : P2 k), (P2.liesOn k P₁ (PascalInscribedHexagon.side k H 0) ∧ P2.liesOn k P₁ (PascalInscribedHexagon.side k H 3)) ∧ (P2.liesOn k P₂ (PascalInscribedHexagon.side k H 1) ∧ P2.liesOn k P₂ (PascalInscribedHexagon.side k H 4)) ∧ (P2.liesOn k P₃ (PascalInscribedHexagon.side k H 2) ∧ P2.liesOn k P₃ (PascalInscribedHexagon.side k H 5)) ∧ P_cubic ≠ 0 ∧ P_cubic.IsHomogeneous 3 ∧ (∀ (i : Fin 6), P2.liesOn k (H.vertex i) P_cubic) ∧ P2.liesOn k p₇ H.conic.poly ∧ (∀ (i : Fin 6), p₇ ≠ H.vertex i) ∧ P2.liesOn k p₇ P_cubic ∧ P2.liesOn k P₁ P_cubic ∧ P2.liesOn k P₂ P_cubic ∧ P2.liesOn k P₃ P_cubic ∧ ¬P2.liesOn k P₁ H.conic.poly ∧ ¬P2.liesOn k P₂ H.conic.poly ∧ ¬P2.liesOn k P₃ H.conic.poly

Auxiliary construction underlying the proof of Pascal's theorem: from a hexagon inscribed in an irreducible conic, exhibit the three opposite-side intersection points, a cubic vanishing on the six vertices and the three intersection points, and an additional seventh conic point witnessing the divisibility argument.

theorem Lec5BezoutPascal.pascal_hexagon_theorem (k : Type u_1) [Field k] [IsAlgClosed k] (H : PascalInscribedHexagon k) (hconic_irred : Irreducible H.conic.poly) : ∃ (P₁ : P2 k) (P₂ : P2 k) (P₃ : P2 k), (P2.liesOn k P₁ (PascalInscribedHexagon.side k H 0) ∧ P2.liesOn k P₁ (PascalInscribedHexagon.side k H 3)) ∧ (P2.liesOn k P₂ (PascalInscribedHexagon.side k H 1) ∧ P2.liesOn k P₂ (PascalInscribedHexagon.side k H 4)) ∧ (P2.liesOn k P₃ (PascalInscribedHexagon.side k H 2) ∧ P2.liesOn k P₃ (PascalInscribedHexagon.side k H 5)) ∧ P2Collinear k P₁ P₂ P₃

Lecture 5, Theorem 5.3 (Pascal): given a hexagon inscribed in an irreducible conic, the three intersection points of opposite sides are collinear.