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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import Mathlib
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Projective
noncomputable section
open MvPolynomial
variable {K σ : Type*} [Field K]
theorem span_isRadical_of_totalDegree_le_one (p : MvPolynomial σ K)
(hp : p.totalDegree ≤ 1) : (Ideal.span ({p} : Set (MvPolynomial σ K))).IsRadical := by
by_cases h0 : p.totalDegree = 0
· rw [(totalDegree_eq_zero_iff_eq_C (p := p)).mp h0]
by_cases hc : p.coeff 0 = 0
· simpa only [hc, C_0, Ideal.span_singleton_zero] using
(Ideal.isRadical_bot : (⊥ : Ideal (MvPolynomial σ K)).IsRadical)
· have hu : IsUnit (C (p.coeff 0) : MvPolynomial σ K) :=
(isUnit_iff_ne_zero.mpr hc).map C
rw [Ideal.span_singleton_eq_top.mpr hu]
exact le_top
· have h1 : p.totalDegree = 1 := by omega
have hirr : Irreducible p := irreducible_of_totalDegree_eq_one h1 (by
intro x hx
apply isUnit_iff_ne_zero.mpr
intro hx0
have hp0 : p = 0 := by
ext d
simpa only [hx0, zero_dvd_iff, AddMonoidAlgebra.coeff_zero, Finsupp.zero_apply] using hx d
simp [hp0] at h1)
exact (Ideal.span_singleton_prime hirr.ne_zero).mpr
(irreducible_iff_prime.mp hirr) |>.isRadical
end
end MaximalSeshadri.Projective
end
end
end
end OAI
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