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import OAI.Algebra.FiniteTensor.JetSolutions
import OAI.Algebra.FiniteTensor.NilpotentSubstitution
import OAI.Algebra.FiniteTensor.OffsetEquations
namespace OAI
/-! Algebraic tensor approximation and the characteristic-zero finite tensor obstruction. -/
noncomputable section
open scoped BigOperators
namespace PD4Tensor.Spreading
noncomputable section
variable {K : Type*} [Field K] [CharZero K] {l m d c : ℕ} {n : Fin l → ℕ}
{e : ((i : Fin l) × Fin (n i)) ≃ Fin d ⊕ Fin c} {active : Fin m ↪ Fin l}
theorem tensor_algebraic_approximation
(M : TensorModel K (fun i=>Fin (n i)) m d c e active)
(W : TensorPrimitives K (fun i=>Fin (n i)) m M)
(hF : ∀ i,IsAlgebraic (MvPolynomial (Fin (n i)) K) (M.F i))
(hb : ∀ i,IsAlgebraic (MvPolynomial (Fin (n (active i))) K) (M.b i)) :
∃ s : JetSolution M W,s.Algebraic := by
classical
obtain ⟨L,hL0,hLeq,hLalg,hLjet⟩ := algebraic_germ_artin (universalLeftEquation M W)
(universalLeftEquation_algebraic M W hF hb)
(fun i=>centerSeries (leftOriginal M W i)) (fun _=>centerSeries_zero _)
(original_solves_left M W) 2
obtain ⟨R,hR0,hReq,hRalg,hRjet⟩ := algebraic_germ_artin (universalRightEquation M W)
(universalRightEquation_algebraic M W hF hb)
(fun i=>centerSeries (rightOriginal M W i)) (fun _=>centerSeries_zero _)
(original_solves_right M W) 2
let s : JetSolution M W := {
left := L
right := R
left_zero := hL0
right_zero := hR0
left_jet := by
intro i j
have h := (mem_jetIdeal_pow_iff_coeff _ 2).mp (hLjet i) (Finsupp.single j 1) (by simp)
simpa only [map_sub,sub_eq_zero] using h
right_jet := by
intro i j
have h := (mem_jetIdeal_pow_iff_coeff _ 2).mp (hRjet i) (Finsupp.single j 1) (by simp)
simpa only [map_sub,sub_eq_zero] using h
left_solve := hLeq
right_solve := hReq }
exact ⟨s,⟨hLalg,hRalg⟩⟩
theorem tensor_algebraic_approximation_target : TensorAlgebraicApproximation := by
intro K _ _ l m d c n e active M W hF hb
exact tensor_algebraic_approximation M W hF hb
theorem no_formal_tensor_model (hm : 5 ≤ m)
(M : TensorModel K (fun i=>Fin (n i)) m d c e active)
(W : TensorPrimitives K (fun i=>Fin (n i)) m M)
(hF : ∀ i,IsAlgebraic (MvPolynomial (Fin (n i)) K) (M.F i))
(hb : ∀ i,IsAlgebraic (MvPolynomial (Fin (n (active i))) K) (M.b i)) : False := by
obtain ⟨s,hs⟩ := tensor_algebraic_approximation M W hF hb
exact no_algebraic_jet_solution hm hF hb s hs
end
end PD4Tensor.Spreading
namespace PD4Tensor
open scoped BigOperators
variable (K : Type*) [Field K] (m : ℕ)
variable {K m}
variable (K m)
variable {K m}
end PD4Tensor
namespace PD4Tensor.Spreading
noncomputable section
variable {K : Type*} [Field K] {m : ℕ} {n : Fin (2*m) → ℕ}
{active : Fin m ↪ Fin (2*m)} {d : ℕ}
def sourceIndex (h : PD4Tensor.Parametrizations K m n d) :
PD4Tensor.Coordinates m n ≃ Fin d ⊕ Fin (PD4Tensor.N m n-d) :=
Fintype.equivOfCardEq (by
simp only [Fintype.card_sum,Fintype.card_fin]
exact (Nat.add_sub_cancel' h.dimension_le).symm)
def sourceModel (g : PD4Tensor.Germs K m n active)
(h : PD4Tensor.Parametrizations K m n d) (hc : PD4Tensor.TensorConditions g h) :
TensorModel K (fun i=>Fin (n i)) m d (PD4Tensor.N m n-d) (sourceIndex h) active where
F := g.F
b := g.b
F_zero := g.F_zero
b_zero := g.b_zero
H := h.HL
G := h.HR
H_zero := h.HL_zero
G_zero := h.HR_zero
tangent_unit := by
apply isUnit_iff_ne_zero.mpr
exact PD4Tensor.joinedMatrix_det_ne_zero (sourceIndex h) _ _
hc.left_injective hc.right_injective hc.complementary
left_vanishing := hc.left_vanishing
right_vanishing := hc.right_vanishing
left_triple := hc.left_triple
right_triple := hc.right_triple
theorem source_primitives_exist (g : PD4Tensor.Germs K m n active)
(h : PD4Tensor.Parametrizations K m n d) (hc : PD4Tensor.TensorConditions g h) :
Nonempty (TensorPrimitives K (fun i=>Fin (n i)) m (sourceModel g h hc)) :=
exists_tensorPrimitives K (fun i=>Fin (n i)) m (sourceModel g h hc)
end
end PD4Tensor.Spreading
theorem required_finite_tensor_obstruction : PD4Tensor.FiniteTensorObstruction := by
intro K _ _ m hm n active g d h hc
obtain ⟨W⟩ := PD4Tensor.Spreading.source_primitives_exist g h hc
exact PD4Tensor.Spreading.no_formal_tensor_model hm
(PD4Tensor.Spreading.sourceModel g h hc) W g.F_algebraic g.b_algebraic
end
end OAI
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