| 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 | |